similar triangles - sample problems.doc

7/28/2019 Similar Triangles - Sample problems.doc

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Similar TrianglesAngles, Sides & similarity ratio

What are similar triangles?

Answer: Similar triangles have the same 'shape' but are just scaleddifferently. Similar triangles have congruent angles and proportional sides.

What is true about the angles of similartriangles?

Answer: They are congruent. as the picture below demonstrates.


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What is true about the sides of similar

triangles?

Answer: Corresponding sides of similar triangles are proportional. Theexample below shows two triangle's with their proportional sides ..

What is the similarity ratio (aka scale

factor)?

Answer: It's the ratio between corresponding sides. In the picture above,the larger triangle's sides are two times the smaller triangles sides so the

scale factor is 2

162 = 32

222 = 44
http://www.mathwarehouse.com/geometry/corresponding-angles-sides.phphttp://www.mathwarehouse.com/geometry/corresponding-angles-sides.php


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252 = 50

Notation: ABC~XYZmeans that "ABC is similar to XYZ"

How do you find the similarity ratio?

Answer: Match up any pair ofcorresponding sides and set up a ratio.

That's it!Example

IfABC~WXY, then what is the similarity ratio?

Step 1) Pick a pairof correspondingsides (follow theletters )

AB and WX are corresponding.

Follow the letters: ABC ~ WXY

Step 2) Substituteside lengths intoproportion

ABWX=721

Step 3) Simplify (ifnecessary)

721=13


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Why is the following problem unsolvable?

If JKL ~ XYZ, LJ = 22 ,JK = 20 and YZ = 30, what is the similarityratio?

Answer: You are not given a single pair ofcorresponding sides so youcannot find the similarity ratio.Remember How to Find corresponding sides

Corresponding sides follow the same letter order as the triangle name so

YZ ofXYZcorresponds with side KL ofJKL

JK ofJKL corresponds with side XY ofXYZ

LJ ofJKL corresponds with side ZX ofXYZ


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Below is a picture of what these two triangles could look like

Practice Problems

Problem 1 If ABC ~ ADE , AB = 20 and AD = 30, what is the similarityratio?


AB and AD are corresponding based on theletters of the triangle names

ABC ~ ADE

Step 2) Substituteside lengths intoproportion

ABAD=2030

Step 3) Simplify (ifnecessary)

2030=23


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Part B) If EA = 33, how long is CA?

EA and CA are corresponding sides (ABC ~

ADE)

Since the sides of similar triangles are proportional, just set up a proportioninvolving these two sides and the similarity ratio and solve.

EACA=3233CA=32CA3=233CA3=66CA=663=22

DE = 27, how long is BC?

EA and AC are corresponding sides ( ABC ~ ADE)Since the sides of similar triangles are proportional, just set up a proportion

involving these two sides and the similarity ratio and solve.DEBC=3227CA=32CA3=227CA3=54CA=543=18

Problem 3) Use your knowledge of similar triangles to find the side lengthsbelow.


HY and HI are corresponding sides

HYZ~ HIY

Step 2) Substituteside lengths into

HYHI=812(You could, of course, have flipped this fractionif youwanted to put HI in the numerator HIHY )
http://www.mathwarehouse.com/geometry/corresponding-angles-sides.phphttp://www.mathwarehouse.com/geometry/corresponding-angles-sides.phphttp://www.mathwarehouse.com/geometry/corresponding-angles-sides.phphttp://www.mathwarehouse.com/fractions/http://www.mathwarehouse.com/geometry/corresponding-angles-sides.phphttp://www.mathwarehouse.com/geometry/corresponding-angles-sides.phphttp://www.mathwarehouse.com/geometry/corresponding-angles-sides.phphttp://www.mathwarehouse.com/fractions/


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proportion

Step 3) Simplify (if

necessary) 812=23

Step 4) Set upequation involvingratio and a pair ofcorresponding sides

23=YZIJ23=YZ9293=YZYZ=6

Finding ZJ is a bit more tricky . You could use the side splitter short cut .Or you use the steps up above to find the length of HJ ,which is 6 and then

subtract HZ (or 4) from that to get the answer.

Problem 4) Below are two different versions of HYZ and HIJ . The onlydifference between the version is how long the sides are.Only one of these two versions includes a pair of similar triangles.Can you identify which version represents similar triangles?
http://www.mathwarehouse.com/geometry/similar/triangles/side-splitter-theorem.phphttp://www.mathwarehouse.com/geometry/similar/triangles/side-splitter-theorem.php


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Similar Triangles

Two triangles are Similar if the only difference is size (and possibly the need to turn or flip

one around).

These triangles are all similar:
http://www.mathsisfun.com/geometry/similar.htmlhttp://www.mathsisfun.com/geometry/similar.html


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(Equal angles have been marked with the same number of arcs)

Some of them have different sizes and some of them have been turned or flipped.

Similar triangles have:

all their angles equal

corresponding sides have the same ratio

Corresponding Sides

In similar triangles, the sides facing the equal angles are always in the same ratio.

For example:

Triangles Rand S are similar. The equal angles are marked with the same numbers of arcs.

What are the corresponding lengths?

The lengths 7 and a are corresponding (they face the angle marked with one arc)

The lengths 8 and 6.4 are corresponding (they face the angle marked with two arcs)

The lengths 6 and b are corresponding (they face the angle marked with three arcs)


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Calculating the Lengths of Corresponding Sides

It may be possible to calculate lengths we don't know yet. We need to:

Step 1: Find the ratio of corresponding sides in pairs of similar triangles.

Step 2: Use that ratio to find the unknown lengths.

Step 1:

We know all the sides in Triangle R, and

We know the side 6.4 in Triangle S (the other sides we call "a" and "b").

The 6.4 faces the angle marked with two arcs as does the side of length 8 in

triangle R.

So we can match 6.4 with 8, and so the ratio of sides in triangle S to triangle Ris:

6.4 to 8 = 64 : 80 = 4 : 5

Now we know that the lengths of sides in triangle S are all 4/5 times the lengths of

sides in triangle R.

Step 2:

And we can then work out a and b:

a faces the angle with one arc as does the side of length 7 in triangle R.

b faces the angle with three arcs as does the side of length 6 in triangle R.

Therefore:

a = 4/5 7 = 28/5 = 5.6

b = 4/5 6 = 24/5 = 4.8

Done!


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How to tell if triangles are similarAny triangle is defined by six measures (three sides, three angles). But you don't need to know all of themto show that two triangles are similar. Various groups of three will do. Triangles are similar if:

1. AAA (angle angle angle)

All three pairs of corresponding angles are the same.

SeeSimilar Triangles AAA.

2. SSS in same proportion (side side side)

All three pairs of corresponding sides are in the same proportion

SeeSimilar Triangles SSS.

3. SAS (side angle side)

Two pairs of sides in the same proportion and the included angle equal.

SeeSimilar Triangles SAS.

Similar Triangles can have shared partsTwo triangles can be similar, even if they share some elements. In the figure below, the larger triangle

PQR is similar to the smaller one STR. S and T are the midpoints of PR and QR respectively. They share

the vertex R and part of the sides PR and QR. They are similar on the basis of AAA, since thecorresponding angles in each triangle are the same.

Try this Drag any orange dot at a vertex of the outer triangle, the inner triangle will change to remain

similar to it, with each corresponding side half the length of the other.

How To Find if Triangles are Similar

Two triangles are similar if they have:

all their angles equal

corresponding sides are in the same ratio

But we don't have to know all three sides and all three angles ...two or

three out of the six is enough.

There are three ways to find if two triangles are similar: AA, SAS and SSS:

AAAA stands for "angle, angle" and means that the triangles have two of their angles equal.

If two triangles have two of their angles equal, the triangles are similar.

For example, these two triangles are similar:
http://www.mathopenref.com/similaraaa.htmlhttp://www.mathopenref.com/similaraaa.htmlhttp://www.mathopenref.com/similarsss.htmlhttp://www.mathopenref.com/similarsss.htmlhttp://www.mathopenref.com/similarsas.htmlhttp://www.mathopenref.com/similarsas.htmlhttp://www.mathsisfun.com/geometry/triangles-similar.htmlhttp://www.mathopenref.com/similaraaa.htmlhttp://www.mathopenref.com/similarsss.htmlhttp://www.mathopenref.com/similarsas.htmlhttp://www.mathsisfun.com/geometry/triangles-similar.html


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If two of their angles are equal, then the third angle must also be equal, because angles of

a triangle always add to make 180.

In this case the missing angle is 180 - (72 + 35) = 83.

So AA could also be called AAA.

SAS

SAS stands for "side, angle, side" and means that we have two triangles where:

the ratio between two sides is the same as the ratio between another two sides

and we we also know the included angles are equal.

If two triangles have two pairs of sides in the same ratio and the included angles are also

equal, then the triangles are similar.

For example:

In this example we can see that:

one pair of sides is in the ratio of 21 : 14 = 3 : 2

another pair of sides is in the ratio of 15 : 10 = 3 : 2

there is a matching angle of 75 in between them
http://www.mathsisfun.com/proof180deg.htmlhttp://www.mathsisfun.com/proof180deg.htmlhttp://www.mathsisfun.com/proof180deg.htmlhttp://www.mathsisfun.com/proof180deg.html


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So there is enough information to tell us that the two triangles are similar.

Using Trigonometry

We could also use Trigonometry to calculate the other two sides using the Law of Cosines:

In Triangle ABC:a2 = b2 + c2 - 2bc cos A

= 212 + 152 - 2 21 15 Cos75

= 441 + 225 - 630 0.2588...

= 666 - 163.055...

= 502.944...

Therefore a = 502.94 = 22.426...

In Triangle XYZ:x2 = y2 + z2 - 2yz cos X

= 142 + 102 - 2 14 10 Cos75

= 196 + 100 - 280 0.2588...

= 296 - 72.469...

= 223.530...

Therefore x = 223.530... = 14.950...

Now let us check the ratio of those two sides:

a : x = 22.426... : 14.950... = 3 : 2

the same ratio as before!

Note: you could also use the Law of Sines to show that the other two angles are equal.

SSS
http://www.mathsisfun.com/algebra/trigonometry-index.htmlhttp://www.mathsisfun.com/algebra/trig-cosine-law.htmlhttp://www.mathsisfun.com/algebra/trig-sine-law.htmlhttp://www.mathsisfun.com/algebra/trigonometry-index.htmlhttp://www.mathsisfun.com/algebra/trig-cosine-law.htmlhttp://www.mathsisfun.com/algebra/trig-sine-law.html


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SSS stands for "side, side, side" and means that we have two triangles with all three pairsof corresponding sides in the same ratio.

If two triangles have three pairs of sides in the same ratio, then the triangles are similar.

For example:

In this example, the ratios of sides are:

a: x = 6 : 7.5 = 12 : 15 = 4 : 5

b: y = 8 : 10 = 4 : 5

c: z = 4 : 5

These ratios are all equal, so the two triangles are similar.

Using Trigonometry

Using Trigonometry we can show that the two triangles have equal angles by using the Law

of Cosines in each triangle:

In Triangle ABC:cos A = (b + c - a)/2bc

= (8 + 4 - 6)/(2 8 4)

= (64 + 16 - 36)/64

= 44/64

= 0.6875

Therefore Angle A = 46.6
http://www.mathsisfun.com/algebra/trigonometry-index.htmlhttp://www.mathsisfun.com/algebra/trig-cosine-law.htmlhttp://www.mathsisfun.com/algebra/trig-cosine-law.htmlhttp://www.mathsisfun.com/algebra/trigonometry-index.htmlhttp://www.mathsisfun.com/algebra/trig-cosine-law.htmlhttp://www.mathsisfun.com/algebra/trig-cosine-law.html


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In Triangle XYZ:cos X = = (y + z - x)/2yz

= (10 + 5 - 7.5)/(2 10 5)

= (100 + 25 - 56.25)/100

= 68.75/100

= 0.6875

Therefore Angle X = 46.6

So angles A and X are equal!

Similarly you can show that angles B and Y are equal, and angles C and Z are equal.

Congruent

If one shape can become another using Turns, Flips and/or Slides, then the

two shapes are called Congruent:

Rotation Turn!

Reflection Flip!
http://www.mathsisfun.com/geometry/rotation.htmlhttp://www.mathsisfun.com/geometry/reflection.htmlhttp://www.mathsisfun.com/geometry/rotation.htmlhttp://www.mathsisfun.com/geometry/reflection.html


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Translation Slide!

After any of those transformations (turn, flip or slide), the shape stillhasthe same size, area, angles and line lengths.

Examples

These shapes are all Congruent:

Rotated Reflected and Moved Reflected and Rotated

Congruent or Similar?

The two shapes need to be the same size to be congruent.

When you need to resize one shape to make it the same as the other, the shapes are

called Similar.

If you ...Then the shapes

are ...

... only Rotate, Reflect and/orTranslate Congruent

... also need to ResizeSimilar
http://www.mathsisfun.com/geometry/translation.htmlhttp://www.mathsisfun.com/geometry/similar.htmlhttp://www.mathsisfun.com/geometry/translation.htmlhttp://www.mathsisfun.com/geometry/similar.html


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Congruent?Why such a funny word that basically means "equal"? Probably because they

would only be "equal" if laid on top of each other. Anyway it comes from Latincongruere, "to

agree". So the shapes "agree"

similar triangles - sample problems.doc

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