Sec Math II Unit 8 Lesson 3 Class Notes EXIT BACKNEXT Click one of the buttons below or press the enter key

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

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. EXIT BACKNEXT

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EXIT BACKNEXT The following are similar figures. I II

EXIT BACKNEXT The following are non-similar figures. I II

EXIT BACKNEXT 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

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

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

1. PPP Similarity Theorem  3 pairs of proportional sides 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. 2. PAP Similarity Theorem  2 pairs of proportional sides and congruent angles between them 3. AA Similarity Theorem  2 pairs of congruent angles

1. PPP Similarity Theorem  3 pairs of proportional sides A BC E F D  ABC   DFE

2. PAP Similarity Theorem  2 pairs of proportional sides and congruent angles between them G H I L JK  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. G H I L JK  Angles I and J do not fall in between sides GH and HI and sides LK and KJ respectively.

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

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. 87  34  S T U X Y Z m  T = m  X m  S = 180  - (34  + 87  ) m  S = 180   m  S = 59  m  S = m  Z  TSU   XZY 59  34 

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

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

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

EXIT BACKNEXT A B C The sum of the measure of the angles of a triangle is    C   Interior Angles of Triangles

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

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

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

Q: Can these triangles be similar? EXIT BACKNEXT

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

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

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

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

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EXIT BACKNEXT A B C P Q R10 6 c 5 4d In the figure, the two triangles are similar. What are c and d ?

EXIT BACKNEXT A B C P Q R10 6 c 5 4d In the figure, the two triangles are similar. What are c and d ?

EXIT BACKNEXT Sometimes we need to measure a distance indirectly. A common method of indirect measurement is the use of similar triangles. h

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

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

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 = x 2 7 =

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

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

SIZES In this case, the factor is x

SIZES Or you can think of the factor as

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 ?

These triangles differ by a factor of ? 15 3= 5

Determine the length of the unknown side ?

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

Sometimes the factor between 2 figures is not obvious and some calculations are necessary ? =

To find this missing factor, divide 18 by ? =

18 divided by 12 = 1.5

The value of the missing factor is =

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

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

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

70 40 The 40 degree angles are congruent.

70 40 The 70 degree angles are congruent.

The other 70 degree angles are congruent.

Find the length of the missing side ?

This looks messy. Let’s translate the two triangles ?

Now “things” are easier to see ? 6

The common factor between these triangles is ? 6

So the length of the missing side is…?

That’s right! It’s ten!

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. 10 feet 2 feet 6 ft

The tree must be 30 ft tall. Boy, that’s a tall tree! 10 feet 2 feet 6 ft

Similar figures “work” just like equivalent fractions

These numerators and denominators differ by a factor of

Two equivalent fractions are called a proportion

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 ? 

1) Determine the missing side of the triangle 

2) Determine the missing side of the triangle  ?

2) Determine the missing side of the triangle  24

3) Determine the missing sides of the triangle ? 8  ?

3) Determine the missing sides of the triangle  11

4) Determine the height of the lighthouse  ?

4) Determine the height of the lighthouse  32

5) Determine the height of the car  ?

5) Determine the height of the car  7.2