Similarity Chapter 8

8.1 Ratio and Proportion  A Ratio is a comparison of two numbers. o Written in 3 ways oA to B oA / B oA : B  A Proportion is an equation where two or more ratios are equal. o

Properties  Cross Product  If a/b = c/d then ad = bc  Reciprocal Property  If a/b = c/d then b/a = d/c

Geometric Mean  The geometric mean of two positive numbers, a and b, is the positive number x, such that:  The geometric mean of 8 and 18 is 12 because: and because:

Solve

Simplify the Ratios

8.2 Problem Solving with Proportions  Additional Properties  If a / b = c / d, then a / c = b / d  If a / b = c / d, then (a + b) / b = (c + d) / d

Mini-Me and Dr. Evil

Mini Horse and Pony

Cheetah Mother with Babies

Find the width to length ratio on each figure. 16mm 20mm 10cm 7.5cm

Find the missing lengths

8.3 Similar Polygons  When all corresponding angles are congruent and lengths of corresponding sides are proportional, the two polygons are similar.  The symbol ~ is used to indicate similarity.

Scale Factor  If two polygons are similar, then the ratio of the lengths of two corresponding sides is called the scale factor. 16 x 5 3.5

Theorem  If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths. Q L M NO P K R ST

Similarity  Are ABCD and EFGH similar?  What is the scale factor? A B CDG H E F

8.4 Similar Triangles  Angle-Angle (AA) Similarity Postulate:  If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.

Similarity  PQR ~ _____  PQ = QR = RP  20 =  y = ____  x = ____ P R Q L M N y x

Similarity  Are the two triangles similar?

Similarity  Are the two triangles similar? 65 50

8.5 Proving Triangles are similar  Side-Side-Side (SSS) Similarity Theorem  If the lengths of the corresponding sides of two triangles are proportional, then the triangles are similar. A C B P Q R IF: THEN:

Side-Angle-Side  (SAS) Similarity Theorem  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. X Z YN P M IF: and THEN:

Examples  Pg 492 #1-5

8.6 Proportions and similar triangles  Four Proportionality Theorems.

Triangle Proportionality Theorem  If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. Q T R S U IF: THEN:

Converse of the Triangle Proportionality Theorem  If a line divides two sides of a triangle proportionally, then it is parallel to the third side. Q T R S U IF: THEN:

Theorems  If three parallel lines intersect two transversals, then they divide the transversals proportionally.  If r ll s and s ll t and l and m intersect r, s, and t, then. rst l m U V W X Y Z

Theorems  If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides. Ifbisects then A C B D

Examples  Pg 502 #1-5