Passy's World of Mathematics

Similar Triangles Applications

Posted on July 18, 2013 by Passy

Image Source: http://www.howitworksdaily.com

A powerful Zoom lens for a 35mm camera can be very expensive, because it actually contains a number of highly precise glass lenses, which need to be moved by a tiny motor into very exact positions as the camera auto focuses.

The Geometry and Mathematics of these lenses is very involved, and they cannot be simply mass produced and tested by computer robots.

Lots of effort required to manufacture these lenses results in their very high price tags.

Here is a diagram showing how the zoom lens internal arrangement changes as we zoom from 18mmm wide angle to 200mm fully zoomed in:

Image Source: http://www.canon.com

Shown above are some band photographs taken by Passy with a special low light camera.

Unfortunately this camera does not have a zoom lens, and so you need to be right up close to the stage to take good pictures.

A special low light aperture 1.4 zoom lens for taking band photographs has a price tag a bit out of Passy’s current reach.

The light rays passing through a camera lens involves some similar triangles mathematics.

We will do some of this mathematics in the “Bow Tie” examples later in this lesson.

Similar Triangles can also be used to measure the heights of very tall objects such as trees, buildings, and mobile phone towers.

Measuring heights of tall objects is also covered in this lesson.

It is very important that you have done our basic lesson on Similar Triangles before doing the lesson which follows on here.

If you need to go back and look at Basic Similar Triangles, then click the link below:

Similar Shapes and Similar Triangles

Bow Tie Triangles

In the above setup for a camera lens, we have a “Bow Tie” shaped pair of Similar Triangles.

Note that when light passes through a camera lens the original image ends up upside down or “inverted”.

This is why cameras have a mirror inside them to put the image right way up so we can view it while taking the photo.

It is very important that this mirror is kept spotlessly clean when changing lenses on a 35mmm camera, and we must be careful never to touch it with our fingers.

Bow Tie Example 1A

The diagram below shows the triangles from our camera lens diagram, with some measured values labelled onto it.

We have used two of the the measurements to work out the “Scale Factor”.

Once we have the S.F. we can then easily work out our missing value.

We do not have to use the Scale Factor method to work out this question.

Instead, we can use the Ratios Cross Multiplying Method, as shown in “Example 1B” below.

It is up to you as to which method you want to use. Both methods give the same correct answer.

Bow Tie Example 1B

In this example we first locate our two pairs of matching sides on the given diagram below.

We then set them up as matching ratios, and use the ratios cross multiplying method to get our answer.

Bow Tie Example 2

Here is another example where we are working with “Bow Tie” Similar Triangles.

In the above example we have used the Scale Factor Method.

This question can also be worked out using cross multiplied ratios, if you prefer to use that method instead.

Video About Bow Tie Questions

The following video shows how to do some example Bow Tie and Ladder Triangle questions.

Using Triangles to Find Height

Similar Triangles can also be used to work out the Heghts of tall objects such as trees, buildings, and towers which are too hard for us to climb and measure with a measuring tape.

Because the sun is shining from a very long way away, it shines down at the same angle on both objects (the person and the tree).

Shadows are formed for both of these objects, because the sun is shining on them at an angle.
Eg. The 2m tall lady makes a 12m long shadow, and the palm tree makes an 84m long shadow.

This results in a pair of similar triangles being formed.

By comparing the lengths of the two shadows, against the two heights, using similar triangles, we can work out the unknown height of the tree.

In the following two examples we show how these types of height questions are drawn as a triangle inside a triangle.

We then use the Scale Factor Method to get our answer for “Example 1A”.

After this, we do the same question using the Cross Multiplying Ratios Method in “Example 1B”.

Finding Height – Example 1A

Finding Height – Example 1B

Finding Height – Example 2

Here is another example of finding height from the shadows, but this time we have a Mobile Phone Tower, and a shorter person with a smaller shadow.

In the above example we have used the Scale Factor Method.

This question can also be worked out using cross multiplied ratios, if you prefer to use that method instead.

Videos About Finding Height

Three and a a half minute video about using shadows to find the height of a tree:

Ten minute video showing a guy actually finding the height of a wall using shadows:

Video showing some algebra x and y problems:

Finding Height Using a Mirror

We can also find the height of a tall object by using line of sight and a mirror, rather than measuring shadows.

This gives a “Bow Tie” type question that we need to solve.

The video at the following link shows an example fo how to do this.

Click here for Video About Finding Height Using a Mirror

River Width Example

Similar Triangles are very useful for indirectly determining the sizes of items which are difficult to measure by hand.

Typical examples include building heights, tree heights, and tower heights.

Similar Triangles can also be used to measure how wide a river or lake is.

Now the instructors could toss a coin to see who ties a rope to themselves, and then swims across the freezing cold water to work out how wide the river is.

However, the following method shown here is much easier, and nobody has to get wet!

It involves each person moving further along the river and measuring exactly how far they have moved from their starting points at A and B.

This is shown in the following diagram:

We can draw in the line of sight from the lady at “E” to the guy on the other side of the river at “C”, which then produces a pair of Similar Triangles.

We can solve these “bow tie” triangles and work out the width of the river as shown below.

(Note that some clipart images from the web were used for the above River Diagrams, and Passy’s World is not claiming any ownership of these cliparts, but only of the mathematical components contained in these examples.)

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Posted in Geometry, Math in the Real World, Triangles | Tagged bow tie similar triangles, bow tie triangles, camera mathematics, finding height of tall buildings, finding height using mirrors, finding height using proprtion, finding height using shadows, finding height using triangles, geometric ratios, Geometry, how to work out river width with triangles, lens geometry, ratios, real life triangles, river width similar triangles, river width triangles, similar trianges in the real world, similar triangles, similar triangles applications, solving ratios, solving triangle ratios, triangles, what are similar triangles used for | Leave a comment

Geometry in the Animal Kingdom

Posted on July 12, 2013 by Passy

Image Source: http://scienceblogs.com

The above photo is of the South African “Weaver Bird”.

Weaver Birds are small bright colored yellow birds, which Passy saw first hand while taking a crocodiles tour at St Lucia.

As their name suggests, they create intricate nests out of plant material, carefully threaded and woven into a solid geometrical structure.

The male birds do lots of intricate weaving to build their nest, and also have to search far and wide to obtain the best building materials.

The birds are not taught by their parents how to weave these nests, it is programmed into their genetics, and they simply “know” how to do it.

Their efforts advertise their skill and quality to potential female mates.

By picking the best built nest, a female gets not only the most comfortable home, but some assurance about the genetic standards of the home-maker.

Nature has thereby programmed these birds so that by natural selection the best genes will be carried forward in the species.

Here is a photograph of a very capable male bird, with a well built geometrically symmetrical nest.

Take a look at the sad face on this other bird in the photo below, and his far from perfect nest. The Geometry is awful, and his selection of building materials is definitely sub-standard.

He is never going to “get a girl” with an effort like that!

No wonder he looks so sad.

See if you can pick the winning nest in this lot.

Many animals use Ornamentation for Mating Purposes.

For the male Weaver Bird, it is the nest he builds, and how good the geometrical shape of it is.

For similar examples of Ornamentation, think about the mane on a lion, the antlers on a deer, or the tail of a peacock.

All of these features serve no practical purpose, other than to attract females.

A Lion’s mane would be very hot and uncomfortable in the harsh African sun, the antlers on a deer require it to eat massive amounts of extra grass to grow them, and the tail of a peacock makes it very hard to move around and fly away from its predators.

The Male Peacock Tail

Image Source: http://www.wired.com

With the male Peacock, attracting a mate is all about geometry.

A male with a well shaped even tail, and a sufficient number of strong even markings that are all similar matching shapes, will be the preferred mate of many female birds.

The quality of the tail is a direct product of great genetics, and so the natural selection process will draw females to the males with the best genes.

There is a great article about Male Peacock Feathers at the following link:

http://www.wired.com/wiredscience/2011/04/peacock-mating-feathers/

Crop Circles in the Ocean

The above video shows another interesting story about Geometry and animals mating in Nature.

The interesting geometrical patterns found on the sea floor are not the work of Aliens, or secret Underwater Artists, but are actually made by Puffer Fish as part of their natural mating ritual!

Geometry of Animal Jaws

A Hippopotamus can open up its jaw to nearly a full 180 degrees.

No wonder they are the most dangerous animal in Africa and kill more people there than any other species.

There is also a lot of geometry going on in the jaws and teeth of sharks.

The jaws of Sharks are made of flexible cartilage.

In contrast to most other fish, sharks’ skeletons are made up entirely of cartilage.

The shark’s upper jaws can be dislocated: the whole upper and lower jaw pull out and forward as the shark twists and shakes its head from side to side to bite a chunk out of its prey.

These sharks feed on very large prey: the great white shark eats sea lions and and its prehistoric ancestor the Megalodon is thought to have eaten whales.

These sharks ambush their prey and immobilise them with a bite, then wait for them to die. They are actually delicate feeders and take care not to damage their teeth by biting down too hard on the large bones of their prey.

To keep their teeth sharp, sharks have a battery of them that is continually replaced. This is why they have rows of teeth in their huge mouth. These rows of teeth are also angled backwards so that their prey cannot escape.

It is the combination of their size, their razor-sharp teeth and the element of surprise that makes sharks such deadly predators.

Geometry of Flight

The following sequence of pictures show an African Eagle taking off at the St Lucia River in South Africa.

Birds flap their wings rapidly through a specific angular range when taking off, and they can gain height rapidly.

In an Eagle, the large wings and their geometry allows the bird to then glide through the air at very high speeds.

Our modern planes have fixed wings that are a specific geometric shape which makes the airflow over them create a lower pressure under the wing which in turn makes the wing rise upwards.

Although a real plane has never flapped its wings like a bird to take off, we do have a photoshop representation of what this would look like:

Image Source: http://www.crazywebsite.com

Early attempts at flight tried to copy the Geometry and Motion of birds, but were highly unsuccessful.

Although, some people have had limited success at the famous Melbourne Bird Man Rally on the Yarra River.

Image Source: http://images.3aw.com.au

That bird man does not look confident.

We doubt he is going to soar like an eagle; in fact not even a redbull could give him wings!

Camouflage in Animals

Another interesting aspect of Geometry in animals is the use of stripes or spots of distinctive geometric shapes, which help the animal hide from predators.

The above Zebras were photographed by Passy in the wild at Kruger National Park in South Africa.

These animals were able to use their stripes to camouflage themselves in amongst the dead trees. They also roll in the dirt and have a lot of brown dust on them to further help them blend into the bush. Very different to the clean hose-washed black and white Zebras we see at the Zoo.

The quality of a Leopard’s spots and their geometrical layout serve two purposes: camouflage when they are running fast, as well as genetic indication to females of their suitability as a potential mate.

It was certainly exciting, and indeed very lucky, when Passy got to see the above sleek and elegant leopard during a night safari in Africa.

Perhaps not quite as exciting or sleek as the Leopard, is its very distant cousin: The Leopard Tortoise.

But once again an African animal with beautiful and functional geometrical markings on its hard strong shell.

Here is a crocodile we saw crawling into some long grass by the river as soon as our boat approached.

Notice how effective his camouflage is once his body enters into the grass.

The strong geometric patterns on his body have merged into his surroundings and he is now much harder to see.

Image Source: http://fineartamerica.com

Here is a member of the animal kingdom with some exquisite geometrical ornamentation: the Diamond Back Rattlesnake.

When it is moving through dry rocky areas, its body is camouflaged, and the level of geometric perfection in its diamonds indicates its genetic quality.

Camouflage For New Model Cars

Image Source: http://images.mnn.com

When car companies are testing yet to be released models, they camoufalge them by deliberately using geometric shapes to confuse the human eye.

These shapes can be Triangles of various sizes, warped chequered flag squares, or various other geometrical designs.

Image Source: http://stwot.motortrend.com

In this next example, Maserati have even used a combination of shapes, which join together to render into three dimensionsal objects wound around the vehicle.

Image Source: http://blogo.it

By far our favorite camouflaged car is the Renault shown in the following photos.

The geometrical shapes and their placement make the side panels look all bent out of shape, making it extremely difficult to see what the car really looks like.

Image Source: www.worldcarfans.com

Here are a couple of photos of the above Renault model after it was mass-produced and marketed to the public.

Image Source: http://images.thecarconnection.com

Image Source: http://www.autoplenum.de

That’s it for this lesson on Geometry in the Real World.

As we have seen, Geometry is extremely important in the real world for the successful propagation of animal species with the best genetics, as well as for the safety and/or hunting prowess gained from camouflage.

We have even been able to copy some of nature’s ideas to camouflage new model cars when they are tested.

This stops car design ideas from being stolen by competitors, as well as preventing the public from getting any sneak previews of new models before they are ready to go on sale.

If you ever get a chance to go to Africa make sure you do.

The people, the animals, and the mathematics are awesome!

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Posted in Geometry, Math in the Real World | Tagged applications of geometry, Geometry, geometry and natural selection, geometry in camoufalge, geometry in new car camouflage, geometry in the real world, how animals use geometry, how is geometry used in real life, mathematical camouflage, practical geometry, Real Life Geometry, real word geometry, what is geometry used for | 6 Comments

Similar Triangles

Posted on July 11, 2013 by Passy

Image Source: www.ebay.com

Similar Triangles turn up in the strangest of places, even in Jewellery made from crystals of the gem stone “Tourmaline”.

Images Sourced from Google Images

In this lesson we look at the nature of similar figures, concentrating on Similar Triangles.

Similar objects have the exact same shape but are different in size.

We cover the methods and rules for establishing similarity.

We also work out how to find unknown sides using Similarity Ratios.

If you are not sure about Ratios, we suggest a review of these previous lessons:

http://passyworldofmathematics.com/introduction-to-ratios/

http://passyworldofmathematics.com/finding-ratio-amounts-of-proportions/

We highly recommend that you have also done our Congruent Triangles lesson, before doing Similar Triangles.

http://passyworldofmathematics.com/congruent-triangles/

In the Similar Triangles lesson here, we do not cover composite similar triangle questions, or applications of Similar Triangles, as these are covered in a separate lesson.

Similar Objects

Two or more items are similar to each other if they have the same shape, but are different sizes.

SIMILAR = SAME SHAPE, DIFFERENT SIZE

Scale Factor

The amount by which we increase, or decrease, the size of an object is called the “Scale Factor” or “S.F.”

The following examples of enlarging and reducing the size of a photo illustrate the concept of Scale Factor.

Calculating the Scale Factor

We calculate the SCALE FACTOR by comparing matching sides, using Ratios.

The following video gives a good introduction to Scale Factor, and also shows some real world applications that involve using Ratio Fractions and Cross Multiplying.

If you are not sure about Ratios, and the Cross Multiplication Method, then check out our previous lesson on this at the link below:

http://passyworldofmathematics.com/finding-ratio-amounts-of-proportions/

Checking for Similar Objects

For two objects to be similar, all their measurements must be changed in the exact same Ratio.

The two objects will then be proportional to each other.

In the example below we first enlarge a 4:3 aspect ratio photo to be double its original size.

The beginning small photo, and the ending larger photo are the exact same shape, and are similar objects.

In the second photo enlargement in our example, we have changed from Standard digital camera 4:3 ratio, to Wide Angle 16:9 aspect ratio.

When we do the ratio proportion mathematics, we find that the ratios of the corresponding sides are NOT the same.

In this 4:3 to 16:9 enlargement, we do not have the same shape, we now have a much wider photo, and so the two photos are NOT Similar.

Similar Triangles

Similar Triangles are the exact Same Shape, but are Different Sizes.

In the remainder of this lesson we will be looking at Similar Triangles.

The following example of two similar triangles involves one triangle, and then a second half size copy of the triangle.

For any two similar triangles their angles will be identical.

However, the sides of the second triangle will be either an Enlargement or a Reduction of the sides of the first triangle.

Introductory Video on Similar Triangles

The following video gives a good introduction to Similar Triangles, including some proofs and problem solving.

Determining Similar Triangles

Similar Triangles need to have the exact same shape, which will happen when their angles are all the same.

Similar Triangles are NOT the same size, which means that their matching sides are NOT equal.

Instead their matching sides are “In Proportion”, eg. either Enlarged or Reduced.

Similar Triangle Rules

Like was the case for Congruent Triangles, there are some “shortcut” rules we can use to prove that two triangles are similar.

Eg. We do not have to check that all three angles are equal, or that all three sides are in proportion.

The shortcut Rules for working out if two triangles are similar to each other are a lot like the rules we use for Congruent Triangles.

There are four Rules for Similar Triangles:

Angle Angle Angle or “AAA”, which turns out to really be just the Angle Angle or “AA” Rule

Proportional Side, Proportional Side, Proportional Side or “PPP” or “SSS” Rule

Proportional Sides, Equal Included Angle, Proportional Sides or “PAP” or “SAS” Rule

Right Angle, Proportional Hypotenuses, Proportional Sides or “RHS” Rule.

Here are some summaries of those four rules.

Angle Angle Angle AAA Rule

If all three Angles are the exact same sizes, (but sizes of triangles are different), then the triangles must be similar.

We actually only need two pairs of matching angles the same, because the third pair will automatically match, because the total angle size in any triangle adds up to 180 degrees.

Proportional Sides PPP or SSS Rule

Here at Passy’s World we like to call this the “PPP” rule, rather than the “SSS” Rule, and leave the “SSS” Rule as being for Congruent equal sized triangles only.

The “PPP” Rule is true if all three sides of the two triangles produce the same Scale Factor value.

If all three sides have the same S.F. then the sides are all in proportion and the two triangles are similar.

Proportional Sides with Angle PAP or SAS Rule

We prefer to call this the “PAP” rule, rather than the “SAS” Rule, and leave the “SAS” Rule as being for Congruent equal sized triangles only.

Right Angled Triangles RHS Rule

This rule is a lot like the RHS rule for congruent equal sized triangles.

However, in this similar triangles rule, the hypotenuses and either pair of the two sides are in Proportion to each other, rather than being equal to each other.

Video About Triangles Rules

The following video covers the four Similar Triangles Rules

Similar Triangles in the Real World

Similar Triangles can be found in Nature, in Art and Craft, and in many structures we design and build.

Amazing Similar Triangles are found inside the crystals of the beautifully colored “Tourmaline” gemstone.

Similar Triangles can create striking effects when used in Art and Craft.

Similar Triangles provide reinforced strength and rigidity to structures, as well as greatly reducing the weight of the objects they are used in.

Examples of Using Similar Triangle Rules

In this first example, we are asked to use either AAA or PPP or PAP or RHS to prove that we have Similar Triangles.

It turns out we need to use the “AAA” Rule.

In this second example, we are asked to use either AAA or PPP or PAP or RHS to prove that we have Similar Triangles.

The two triangles have two sides and the included Angles inbetween these two sides.

Therefore we need to used the PAP / SAS Rule.

Examples of Solving Similar Triangles

In this first Example, we need to first prove that the Two Trianges are Similar.

After this we can find the Scale Factor that exists between the two tiangles.

Using the S.F. value, we then find the unknown side.

Using the Scale Factor works well for situation where we have the known side on a small triangle, and we need to find the unknown length of the matching side on a big triangle.

However, if the situation if the other way around, we get fraction Scale Factors such as one third 1/3 to deal with and this makes the mathematical working out a bit fiddly.

Another way to solve similar triangles is to write two rations and then use “cross multiplying”, (or “Cross Products”).

At Passy’s World we prefer to use this “Cross Products” method for solving triangle questions, because it helps avoid dealing with fractions.

In “Example 3B” below, we redo “Example 3A”, but this time use the Cross Products Method to solve our triangle question.

In this final “Example 3C” our question is the other way around, and we have to find an unknown side on the smaller triangle.

We could work out that the Scale Fator is 10/30 and reduce this down to 1/3. But then we get into fractions work to get to the final answer.

Here at Passy’s World, we have found that students have difficulty with fractions, and so we have worked out “Example 3C” using Cross Multiplyling as shown below.

Using Cross Multiplying avoids having to deal with fractions, and we believe that is a good thing.

Videos About Solving Similar Triangles

Here are some videos where unknown sides are found for Similar Triangles

Similarity concepts can also be applied to Quadrilaterals as well as Trinagles.

This is demonstrated very well in the following video by Mr Bill Konst.

Music Video About Similar Triangles

Here is a bit of a humorous rap song about Similar Triangles.

There are some extremely, but a bit way out, mathematical concepts related to triangles covered in this entertaining little story.

Interactive Similar Triangles

The above Online Manipulative, (which can go full screen), allows us to drag corner vertices around and make any kind of shaped pair of similar triangles.

Click the link below to try it out:

http://www.mathopenref.com/similartriangles.html

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Posted in Geometry, Triangles | Tagged aspect ratio, cross multiply, cross products, enlargement, Geometry, how to do similar triangles, identify similar triangles, pap rule, ppp rule, real world similar triangles, reductions, rhs rule, scale factor, scaled triangles, similar shapes, similar triangle rules, similar triangles, similar triangles examples, similar triangles in real life, Similar Triangles Powerpoint, Similar Triangles PPT, similar triangles rap, similar triangles song, similarity, solving triangle ratios, triangle ratios | 7 Comments

Congruent Triangles

Posted on July 6, 2013 by Passy

Image Source: http://resources3.news.com.au

Identical Twins have the exact same size and shape, and we instantly recognise that the two of them are exactly the same.

When two items have the exact same size and shape, we say that they are “Congruent”.

This lesson is all about “Congruent Triangles”, eg. pairs of Triangles which have the exact same size and shape.

Congruent Triangles are an important part of our everyday world, especially for reinforcing many structures.

Here are a typical pair of Congruent Triangles

Two triangles are congruent if they are completely identical.

This means that the matching sides must be the same length and the matching angles must be the same size.

The Identical (eg.”Congruent”) Triangles can be in different positions, (or orientations), and still be the exact same size and shape.

The position of the matching Triangles does not affect the fact that they are identical, or “Congruent”.

Introductory Video About Congruency

The following Video by Mr Bill Konst about Congruence, covers the “SSS Rule for Triangles”, as well as covering Quadrilaterals and some interesting optical illusions.

Shortcut Rules for Congruent Triangles

It turns out that we do not have to check all the sides and angles of two Triangles to work out that they are Congruent.

There are FOUR “Shortcut Rules” for Congruent Triangles that we will be covering in this lesson.

The first of these “Shortcut Rules” is the “Side Side Side”, or “SSS” Rule.

SSS – Side Side Side Rule for Triangles

We can actually use just the three sides to work out if two triangles are congruent.

This is called “SSS” or the “Side Side Side Rule”.

In diagrams, the actual values of the sides are sometimes not given.

Instead we have markers to show where the matching same length sides are on the two triangles.

For these Triangles we can apply the “SSS” rule, as long as we have all three sides matching each other on the two triangles.

Symbols Used in Congruency

There is a special symbol we use to indicate that Triangles are Congruent, that is like an equals sign with an extra line added on top of it.

This triple ine symbol is what we use in Australia; however some other countries use an equals sign with a squiggly “tilda” line added to the top of it.

In addition, Triangles are usually labelled with capital alphabet letters.

When we say the Triangles are Congruent using their letters, we need to make sure the order of the letters matches the path around the two triangles correctly.

This is shown in the following example.

We need to be careful with the labelling when our Triangles are in different positions.

Eg. We need to make sure the order of the letters matches going around the two triangles in the same order of sides. Going A to B to C should be the exact same path as D to E to F.

SAS – Side Angle Side Rule for Triangles

Two Triangles will be congruent if two matching sides have equal lengths and
the angle included by these sides is the same.

The following video covers the “SSS” and “SAS” Rules for Congruent Triangles.

The American teacher doing the videos does not always use the most correct language, but he is enthusiastic and explains his examples well.

AAS – Angle Angle Side Rule for Triangles

Two triangles are congruent if two matching angles are equal and a matching side is equal in length.

There is also an old “ASA” Angle Side Angle Rule; however this has been brought in to be part of the “AAS” Rule.

It is quite okay to use the “ASA” Rule if the order of the items is “ASA” as shown in the above diagram.

However, most people these days just use “AAS”, so that there is one less congruency rule to memorise.

The following “YayMath” 26 minute gives a comprehensive lesson on the ASA and AAS Rules.

RHS – Right Angle Hypotenuse Rule

Any two Right-Angled 90 degree Triangles are congruent if the hypotenuse and one pair of matching sides are equal in length.

This Rule works because of Pythagoras Theorem for 90 Degree Triangles.

Pythagoras Rule means that the missing side lengths have to be equal, so we are indirectly using the “SSS” Rule here.

Why the SSA Rule Does Not Exist

The following video shows why there is not an SSA Rule for congruent triangles.

Triangle Rules Summary

There are four rules that we use to determine if Triangles are congruent: SSS, SAS, AAS, and RHS.

These are shown in the diagram below:

Congruent Triangles Music Videos

Here is a quick little tune by Abbie about the Triangle Rules, mentioning exclusion of the invalid Angle Side Side “Donkey Rule”.

This next one is a heavy metal Parody (sounds a bit like a Joan Jet song):

Congruent Triangles Examples

The following examples show the required working out for demonstrating that a pair of Triangles are identical.

Composite Shapes Examples

The following video shows some more complex examples, where triangle sides are joined together to other triangles and shapes.

In the next two examples, Congruent Triangles are found within the given Geometric Shapes, which allows side lengths to be proven as equal.

Rules for Angles in Parallel Lines are also used, in particular, the following Alternate Interior Angles Rule:

This first example is the classic “Bow Tie” shaped question for joined congruent triangles.

The next example involves two triangles sharing the diagonal of a Parallelogram.

Congruent Triangles Worksheets

The following worksheet has basic multiple choice questions on Congruent Triangles.

(There are answers on the last page of the PDF document).

The following video has an worked example to get you started.

Click the link below to do this worksheet:

Basic Level Worksheet

The following worksheet has medium level of difficulty multiple choice questions on Congruent Triangles.

(There are answers on the last page of the PDF document).

The following video has an worked example to get you started.

Click the link below to do this worksheet:

Medium Level Worksheet

This last worksheet has challeging multiple choice questions on Congruent Triangles.

(There are answers on the last page of the PDF document).

The following video has an worked example to get you started.

Click the link below to do this worksheet:

Challenging Worksheet

Congruent Triangles Games

The above game is a matching game with several levels of difficulty.

It is more like a Quiz than a Game, but will help you learn congruent triangles.

Click the link below to play this game.

http://www.mangahigh.com/en_au/maths_games/shape/congruence/congruent_triangles

This next game involves dropping the shapes into the correct boxes using the arrow keys.

It also contains multiple choice questions that need to be answered.

Click the link below to play this game.

http://www.classzone.com/books/geometry_concepts/page_build2.cfm?CFID=15754374&ch=5&id=game

The following is an activity where we get to build congruent triangles based on the congruency rule we pick to work with.

Click the link below to play this game.

http://illuminations.nctm.org/ActivityDetail.aspx?ID=4

This final game is a Jeapordy Game, but is very slow to load up.

Click the link below to play this game.

http://www.superteachertools.com/jeopardyx/jeopardy-review-game.php?gamefile=1322685871

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Posted in Geometry, Triangles | Tagged congruency, congruent rules, congruent shapes, congruent triangles, Congruent Triangles Game, Congruent Triangles Powerpoint, Congruent Triangles PPT, Geometry, how to do congruent triangles, identical triangles, identify congruent triangles, real world congruent shapes, real world congruent triangles, triangles, what does congruent mean | 8 Comments

Factorising With Common Factors

Posted on July 1, 2013 by Passy

The answer to this question is quite simple:

Looking at the two photos, it is quite easy for a human to work out which person is in both pictures.

But imagine the Maths that would be needed, (coupled with some facial recognition software), for a computerised device to work this out.

Actually the program required is not too complicated. It could use a Database of Facial Images, a lookup table, and some mathematics in the form of Sort/Match array processing code.

But this is turning into a High School IT lesson, so let’s get back to the Maths!

In this lesson we will be looking at Algebra Terms in expressions, and working out if the separate terms have anything in common.

Once we have worked out the “Common Factor”, it is then a fairly simple task to rewrite our original expression in “Factor Form”, (eg. a Form which contains brackets).

Prime Factors and Factor Trees

Before we can do the Common Factor work for this lesson, we first need to review “Prime Factors”.

In the images shown here we are building “Factor Trees” to work out the Prime Factors.

The Tree shapes are actually upside down in these diagrams, and the ends of each branch are called the “leaves” and this is where our Prime Number factors end up.

We combine together the leaves on the ends of the branches, and this gives us our set of Prime Factors.

Watch the videos in the following section if you do not already know how to make “Prime Factors” for numbers.

Prime Factors Videos

And finally watch this great music song video from “Math Rocks!”

Expanded Form and Factor Form

When we expanded brackets using the Distributive Rule, we created an answer containing no brackets which is called “Expanded Form”.

Expanded Form is useful when we wnat to solve an Equation, or draw a Graph on an X-Y Grid.

If you need to review expanding brackets, then check out our previous lesson at the link below:

http://passyworldofmathematics.com/expanding-brackets-using-distributive-rule/

In this lesson we are doing the reverse process of Expanding, which is called “Factorisation”.

Factorising results in us getting an Expression which contains brackets.

Factorised Expressions are used for locating the horizontal X-Axis cross-over points on Graphs.

Factorising Steps

The steps we follow to convert an Expanded expression into a Factorised expression are as follows.

Common Factor Examples

The following examples show how we do Factorisation to create answers that have a single bracket and are in Factor Form.

Common Factors Involving Exponents

When our initial terms contain variable letters and exponents, we need to expand out the exponents, and then see what letters we have in common.

The following examples show this; where numbers and letters are both broken down as low as they can go, and then the common factors identified.

Common Factors Involving a Negative Sign

In the example below, we have a negative sign on both items, which means the negative sign is a Common Factor.

We write the negative sign as a -1 when we include it in the expanded factors.

Videos About Factorising

The following video shows how to find the Highest Common Factor (HCF), or Greatest Common Factor (GCF), using Factor Trees.

This next video shows how to do the greatest common factor a faster way than using Prime Factors.

Common Factor Games

“Fruit Shoot” is a fun game where you have to work out what the biggest Common Factor is, and then use the mouse to move onto the correct piece of fruit and click on it.

Click the following link to play this fun game:

http://www.sheppardsoftware.com/mathgames/fractions/GreatestCommonFactor.htm

In this next Game, we slide the bottom pink boxes to their correct position on the question grid.

Click the following link to play this fun game:

http://www.oswego.org/ocsd-web/match/dragflip.asp?filename=slanegcf

If you enjoyed this lesson, why not get a free subscription to our website.
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Go to the subscribe area on the right hand sidebar, fill in your email address and then click the “Subscribe” button.

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Help us to maintain this free service and keep it growing.

Donate any amount from $2 upwards through PayPal by clicking the PayPal image below. Thank you!

Enjoy,
Passy

Posted in Algebra | Tagged Algebra, CF, common factor, common factor powerpoint, common factors, expanded form, extracting factors, factor form, factor tree, factorising, factorising single brackets, factors games, GCF, GCF games, getting common factor, greatest common factor, greatest common factor games, HCF, highest common factor, highest common factor games, how to do common factor, how to factorise, prime factors, prime numbers, reverse distributive law, taking out the common factor, what is a common factor | Leave a comment

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