Similarity of Triangles Course 3 5-5 Class Notes Click one of the buttons below or press the enter key BACK NEXT EXIT

In geometry, two polygons are similar when one is a replica (scale model) of the other. BACK NEXT EXIT

Consider Dr. Evil and Mini Me from Mike Meyers’ hit movie Austin Powers. Mini Me is supposed to be an exact replica of Dr. Evil. BACK NEXT EXIT

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The following are similar figures. II BACK NEXT EXIT

The following are non-similar figures. II BACK NEXT EXIT

Feefee the mother cat, lost her daughters, would you please help her to find her daughters. Her daughters have the similar footprint with their mother. Feefee’s footprint BACK NEXT EXIT

Which of the following is similar to the above triangle? 1. Which of the following is similar to the above triangle? B A C BACK NEXT EXIT

Similar triangles are triangles that have the same shape but not necessarily the same size. D F E ABC  DEF When we say that triangles are similar there are several repercussions that come from it. A  D AB DE BC EF AC DF = = B  E C  F

Six of those statements are true as a result of the similarity of the two triangles. However, if we need to prove that a pair of triangles are similar how many of those statements do we need? Because we are working with triangles and the measure of the angles and sides are dependent on each other. We do not need all six. There are three special combinations that we can use to prove similarity of triangles. 1. PPP Similarity Theorem  3 pairs of proportional sides 2. PAP Similarity Theorem  2 pairs of proportional sides and congruent angles between them 3. AA Similarity Theorem  2 pairs of congruent angles

ABC  DFE E F D 1. PPP Similarity Theorem  3 pairs of proportional sides 9.6 10.4 A B C 5 13 12 4 ABC  DFE

GHI  LKJ mH = mK 2. PAP Similarity Theorem  2 pairs of proportional sides and congruent angles between them L J K 7.5 G H I 5 70 70 7 10.5 mH = mK GHI  LKJ

The PAP Similarity Theorem does not work unless the congruent angles fall between the proportional sides. For example, if we have the situation that is shown in the diagram below, we cannot state that the triangles are similar. We do not have the information that we need. L J K 7.5 G H I 5 50 7 50 10.5 Angles I and J do not fall in between sides GH and HI and sides LK and KJ respectively.

MNO  QRP mN = mR mO = mP 3. AA Similarity Theorem  2 pairs of congruent angles Q P R M N O 70 50 50 70 mN = mR MNO  QRP mO = mP

TSU  XZY mT = mX mS = mZ It is possible for two triangles to be similar when they have 2 pairs of angles given but only one of those given pairs are congruent. S T U X Y Z 34 34 34 34 59 59 mT = mX 87 59 mS = mZ mS = 180- (34 + 87) TSU  XZY mS = 180- 121 mS = 59

Note: One triangle is a scale model of the other triangle. BACK NEXT EXIT

How do we know if two triangles are similar or proportional? BACK NEXT EXIT

Triangles are similar (~) if corresponding angles are equal and the ratios of the lengths of corresponding sides are equal. BACK NEXT EXIT

The sum of the measure of the angles of a triangle is 1800. Interior Angles of Triangles A B C The sum of the measure of the angles of a triangle is 1800. Ð A + Ð B + ÐC =1800 BACK NEXT EXIT

Determine whether the pair of triangles is similar. Justify your answer. Answer: Since the corresponding angles have equal measures, the triangles are similar. Example 6-1b

If the product of the extremes equals the product of the means then a proportion exists. BACK NEXT EXIT

This tells us that  ABC and  XYZ are similar and proportional.   This tells us that  ABC and  XYZ are similar and proportional. BACK NEXT EXIT

Q: Can these triangles be similar? BACK NEXT EXIT

Answer—Yes, right triangles can also be similar but use the criteria. BACK NEXT EXIT

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Do we have equality? This tells us our triangles are not similar. You can’t have two different scaling factors! BACK NEXT EXIT

If we are given that two triangles are similar or proportional what can we determine about the triangles? BACK NEXT EXIT

The two triangles below are known to be similar, determine the missing value X. BACK NEXT EXIT

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In the figure, the two triangles are similar. What are c and d ? B C P Q R 10 6 c 5 4 d BACK NEXT EXIT

In the figure, the two triangles are similar. What are c and d ? B C P Q R 10 6 c 5 4 d BACK NEXT EXIT

Sometimes we need to measure a distance indirectly Sometimes we need to measure a distance indirectly. A common method of indirect measurement is the use of similar triangles. h 6 17 102 BACK NEXT EXIT

MODELING A REAL-LIFE PROBLEM Error Analysis GEOMETRY CONNECTION Two students are visiting the mysterious statues on Easter Island in the South Pacific. To find the heights of two statues that are too tall to measure, they tried a technique involving proportions. They measured the shadow lengths of the statues at 2:00 P.M. and again at 3:00 P.M. 3:00 2:00

a b a b 2:00 3:00 a = b a = b a = b a = b a = b a = b Error Analysis SOLUTION They let a and b represent the heights of the two statues. Because the ratios of corresponding sides of similar triangles are equal, the students wrote the following two equations. 27 a 18 b 30 a 20 b 2:00 3:00 a 27 = b 18 a 30 = b 20 a = 27 18 b a = 30 20 b 30 ft 27 ft 18 ft 20 ft a = 3 2 b a = 3 2 b

Draw Similar Rectangles ABCD and EFGH whose lengths and widths are 16 and 12 and 12 and 9 respectively.

12 16 9 12

Two triangles are called “similar” if their corresponding angles have the same measure.      

a A b B c C Two triangles are called “similar” if their corresponding angles have the same measure.  Ratios of corresponding sides are equal. C A  a c     b B a A b B c C = =

Mary is 5 ft 6 inches tall. She casts a 2 foot shadow. The tree casts a 7 foot shadow. How tall is the tree?

Mary is 5 ft 6 inches tall. She casts a 2 foot shadow. The tree casts a 7 foot shadow. How tall is the tree? Mary’s height Tree’s height Mary’s shadow Tree’s shadow = x 5.5 2 7

Mary is 5 ft 6 inches tall. She casts a 2 foot shadow. The tree casts a 7 foot shadow. How tall is the tree? 5.5 x 2 7 = Mary’s height Tree’s height Mary’s shadow Tree’s shadow = x 5.5 2 7

5.5 x 2 7 = 7 ( 5.5 ) = 2 x 38.5 = 2 x x = 19.25 The height of the tree is 19.25 feet

Find the missing measures if the pair of triangles is similar. Corresponding sides of similar triangles are proportional. and Example 6-2b

Find the cross products. Divide each side by 4. Answer: The missing measure is 7.5. Example 6-2b

Find the missing measures if each pair of triangles is similar. a. Answer: The missing measures are 18 and 42. Example 6-2c

Find the missing measures if each pair of triangles is similar. b. Answer: The missing measure is 5.25. Example 6-2c

Shadows Richard is standing next to the General Sherman Giant Sequoia three in Sequoia National Park. The shadow of the tree is 22.5 meters, and Richard’s shadow is 53.6 centimeters. If Richard’s height is 2 meters, how tall is the tree? Since the length of the shadow of the tree and Richard’s height are given in meters, convert the length of Richard’s shadow to meters. Example 6-3a

Let the height of the tree. Simplify. Let the height of the tree. Richard’s shadow Tree’s shadow Richard’s height Tree’s height Cross products Answer: The tree is about 84 meters tall. Example 6-3a

Answer: The length of Trudie’s shadow is about 0.98 meter. Tourism Trudie is standing next to the Eiffel Tower in France. The height of the Eiffel Tower is 317 meters and casts a shadow of 155 meters. If Trudie’s height is 2 meters, how long is her shadow? Answer: The length of Trudie’s shadow is about 0.98 meter. Example 6-3b

Congruent Figures In order to be congruent, two figures must be the same size and same shape.

Similar Figures Similar figures must be the same shape, but their sizes may be different.

Similar Figures This is the symbol that means “similar.” These figures are the same shape but different sizes.

SIZES Although the size of the two shapes can be different, the sizes of the two shapes must differ by a factor. 4 2 6 6 3 3 1 2

In this case, the factor is x 2. SIZES In this case, the factor is x 2. 4 2 6 6 3 3 2 1

Or you can think of the factor as 2. SIZES Or you can think of the factor as 2. 4 2 6 6 3 3 2 1

When you have a photograph enlarged, you make a similar photograph. Enlargements When you have a photograph enlarged, you make a similar photograph. X 3

Reductions A photograph can also be shrunk to produce a slide. 4

Determine the length of the unknown side. 15 12 ? 4 3 9

These triangles differ by a factor of 3. 15 3= 5 15 12 ? 4 3 9

Determine the length of the unknown side. ? 2 24 4

These dodecagons differ by a factor of 6. ? 2 x 6 = 12 2 24 4

Sometimes the factor between 2 figures is not obvious and some calculations are necessary. 15 12 8 10 18 12 ? =

To find this missing factor, divide 18 by 12. 15 12 8 10 18 12 ? =

18 divided by 12 = 1.5

The value of the missing factor is 1.5. 15 12 8 10 18 12 1.5 =

When changing the size of a figure, will the angles of the figure also change? 40 70 ? ? 70

Nope! Remember, the sum of all 3 angles in a triangle MUST add to 180 degrees. If the size of the angles were increased, the sum would exceed 180 degrees. 40 40 70 70 70 70

We can verify this fact by placing the smaller triangle inside the larger triangle. 40 40 70 70 70 70

The 40 degree angles are congruent. 70 70 70 70

The 70 degree angles are congruent. 40 40 70 70 70 70 70

The other 70 degree angles are congruent. 4 40 70 70 70 70 70

Find the length of the missing side. 50 ? 30 6 40 8

This looks messy. Let’s translate the two triangles. 50 ? 30 6 40 8

Now “things” are easier to see. 50 30 ? 6 40 8

The common factor between these triangles is 5. 50 30 ? 6 40 8

So the length of the missing side is…?

That’s right! It’s ten! 50 30 10 6 40 8

Similarity is used to answer real life questions. Suppose that you wanted to find the height of this tree.

Unfortunately all that you have is a tape measure, and you are too short to reach the top of the tree.

You can measure the length of the tree’s shadow. 10 feet

Then, measure the length of your shadow. 10 feet 2 feet

If you know how tall you are, then you can determine how tall the tree is. 6 ft 10 feet 2 feet

The tree must be 30 ft tall. Boy, that’s a tall tree! 10 feet 2 feet

Similar figures “work” just like equivalent fractions. 30 5 11 66

These numerators and denominators differ by a factor of 6. 30 6 5 6 11 66

Two equivalent fractions are called a proportion. 30 5 11 66

Similar Figures So, similar figures are two figures that are the same shape and whose sides are proportional.

Practice Time!

1) Determine the missing side of the triangle. ? 9 5  3 4 12

1) Determine the missing side of the triangle. 15 9 5  3 4 12

2) Determine the missing side of the triangle. 36 36 6 6  4 ?

2) Determine the missing side of the triangle. 36 36 6 6  4 24

3) Determine the missing sides of the triangle. 39 ?  33 ? 8 24

3) Determine the missing sides of the triangle. 39 13  33 11 8 24

4) Determine the height of the lighthouse.  ? 8 2.5 10

4) Determine the height of the lighthouse.  32 8 2.5 10

5) Determine the height of the car.  ? 3 5 12

5) Determine the height of the car. 7.2  3 5 12

Similarity of Triangles Algebra 1 Honors 4-2 Click one of the buttons below or press the enter key BACK NEXT EXIT