Chapter 8 Similarity

Chapter 8 Objectives Define a ratio Manipulate proportions Use proportions to solve geometric situations Calculate geometric mean Identify similar polygons Prove triangles are similar Use properties of similar triangles Perform dilations

Lesson 8.1 Ratio and Proportion

Ratio If a and b are two quantities measured in the same units, then the ratio of a to b is a/b.a/b. – It can also be written as a:b. A ratio is a fraction, so the denominator cannot be zero. Ratios should always be written in simplified form. – 5 / 10  1/21/2

Simplifying Ratios Not only should ratios be in simplified form, but they must also be in the same units! Example 1 – 12 cm / 4 m Make sure they units are the same before simplifying the numbers! – 12 cm / 4 m(100 cm) = 12 cm / 400 cm  3 cm / 100 cm Some info to keep in mind when changing units – 100 cm = 1 m – 1000 m = 1 km – 12 in = 1 ft – 3 ft = 1 yd – 5280 ft = 1 mile – 16 oz = 1 lb

Example 2 Sometimes you may be given a problem that states the ratios of side lengths or angle measures. The ratio of the measures of the angles in a triangle are 1:2:3. Find the measures of all three angles. – You must set one of the angles equal to x and adjust the other according to the ratio. x 2x 3x x + 2x + 3x = 180 o 6x = 180 o x = o 90 o

Proportion An equation that has two ratios equal to each other is called a proportion. – A proportion can be broken down into two parts. Extremes – Which is the numerator of the first ratio and the denominator of the second ratio Means – Which is the denominator of the first and numerator of the second. b ca d =

Properties Of Proportions Cross Product Property – The product of the extremes equals the product of the means. Also known as cross- multiplying. Reciprocal Property – Taking the reciprocal of the entire proportion creates an equivalent proportion. b ca d = ad=bc b ca d = a db c =

Solving Proportions To solve a proportion, you must use the cross product property. So multiply the extremes together and set them equal to the means. b ca d = ad=bc

Example 3 Solve the following proportions using the Cross Product Property I. II. 24 bb + 3 = 23 9x =

Lesson 8.3 Similar Polygons

Similarity of Polygons Two polygons are similar when the following two conditions exist – Corresponding angles are congruent – Correspondng sides are proportional Means that all side fit the same ratio. The symbol for similarity is –~–~ ABCD ~ EFGH – This is called a similarity statement.

Proportional Statements Proportional statements are written by identifying all ratios of corresponding sides of the polygons. – Assume that square ABCD  EFGH A B D C E F H G AB / EF = BC / FG = CD / GH = AD / EH

Scale Factor Since all the ratios should be equivalent to each other, they form what is called the scale factor. – We represent scale factor with the letter k. This is most easily found by find the ratio of one pair of corresponding side lengths. – Be sure you know the polygons are similar. A B D C E F H G k = 20 / 5 k =

Theorem 8.1: Similar Perimeters If two polygons are similar, then the ratio of their perimeters is equal to the ratio of their side lengths. – This means that if you can find the ratio of one pair of corresponding sides, that is the same ratio for the perimeters.

Lesson 8.4 Similar Triangles

Postulate 25: Angle-Angle Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.

Example 4 Find the length of side RT > > NP TR Q  RQT   PQN Vertical Angles  QNP   QTR Alternate Interior Angles  NPQ   TRQ AA Similarity Be sure that  NPQ   TRQ. To do so try to use AA to find two angles that are congruent to each other. Hint: Parallel Line Postulates x / 18 = 6 / 12 12x = (18)(6) 12x = 108 x = 9

Lesson 8.5 Proving Triangles are Similar

Theorem 8.2: Side-Side-Side Similarity If the corresponding sides of two triangles are proportional, then the triangles are similar. – Your job is to verify that all corresponding sides fit the same exact ratio!

Theorem 8.3: Side-Angle-Side Similarity If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar. – Your task is to verify that two sides fit the same exact ratio and the angles between those two sides are congruent!

Example 5 Identify the similar triangles, if any. If so, explain how you know they are similar and write a similarity statement. A B C D E F G H I  ABC   DEF, by SSS Similarity

Using Theorems 8.2 and 8.3 These theorems share the abbreviations with those from proving triangles congruent in chapter 4. – SSS – SAS So you now must be more specific – SSS Congruence – SSS Similarity – SAS Congruence – SAS Similarity You chose based on what are you trying to show? – Congruence – Similarity

Lesson 8.6 Proportions and Similar Triangles

Triangle Proportionality Theorem 8.4: Triangle Proportionality – If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. Theorem 8.5: Converse of Triangle Proportionality – If a line divides two sides proportionally, then it is parallel to the third side. Q S R T U If TU // QS, then RT / TQ = RU / US. If RT / TQ = RU / US, then TU // QS.

Example 6 Determine what they are asking for – If they are asking to solve for x Make sure you know the sides are parallel! – If they are asking if the sides are parallel Make sure you know the ratio of sides lengths are the same. Q S R T U x / 4 = x / 2 4x = 20 x = 5 x / 2 = 10 / 4 4x = (10)(2) 4x = 20 x = 5

Theorem 8.6: Proportional Transversals If three parallel lines intersect two transversals, then they divide the transversals proportionally.

Lesson 8.7 Dilations

Dilation A dilation is a transformation with the following properties – If point P is not at the center C, then the image P’ lies on ray CP. – If point P is at the center, then P = P’. A dilation is something that will increase or decrease the size of the figure while still maintaining similarity. C P P’

Scale Factor of a Dilation The scale factor of a dilation is found by the following – k = CP’ / CP k stands for scale factor It is basically the distance from the center to the image divided by the distance from the center to the pre-image. C P P’ 3 12 k = 12 / 3 k = 4

Reduction or Enlargement A reduction is when the image is smaller than the pre-image. The scale factor will be a number between 0 and 1. – 0 < k < 1 An enlargement is when the image is larger that the pre-image. The scale factor will be a number greater than 1. – k > 1 C C

Scale Factor with Coordinates Center at the Origin When applying the scale factor to a set of coordinates, simply distribute to both the x and y values of each coordinate. Example – Perform the following dilation for point P k = 3, P(2,7) – P’(32,37) » P’(6,21)