Similarity. Similar Polygons
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- Amberly Carr
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1 Similarity Similar Polygons 1
2 MAKING CONNECTIONS Dilating a figure produces a figure that is the same as the original figure, but a different. Like motions, dilations preserve measures. Unlike rigid motions, dilations do not preserve the of line segments. Instead, they produce a figure with sides that are to the sides of the pre-image. So, the original figure and its image are figures. A polygon is a polygon in which all sides have the length and all angles have the same. Any two regular polygons of the same type having the same number of sides - are similar to each other. The symbol for similar is. 2
3 MAKING CONNECTIONS Dilating a figure produces a figure that is the same as the original figure, but a different. Like motions, dilations preserve measures. Unlike rigid motions, dilations do not preserve the of line segments. Instead, they produce a figure with sides that are to the sides of the pre-image. So, the original figure and its image are similar regular same figures. size angle length proportional resulting A polygon is a polygon in which all sides have the length and all angles have the same. Any two regular polygons of the same type having the same number of sides - always are similar to each other. ~ The symbol for similar is. shape rigid measure 3
4 Characteristics of Similar Polygons 1. Corresponding angles are. 2. Corresponding sides are. Similarity Statement ABC ~ DEF
5 EXS: Are the pairs of figures similar? EXPLAIN. Corresponding Sides : Corresponding Sides 1) 2) 3) 4)
6 EXS: Are the pairs of figures similar? EXPLAIN. Corresponding Sides : Corresponding Sides 1) 2) 3) 4)
7 Setting Up the Ratio of Similar Polygons 1) Find the Scale Factor. 2) Set the scale factor equal to a ratio containing the missing side to form a proportion. 3) Solve for the variable. 8/24/2018
8 EXS: Each pair of polygons is similar. Write a proportion to find each missing side. Solve for x. 1) 2) 3)
9 EXS: Each pair of polygons is similar. Write a proportion to find each missing side. Solve for x. 1) 2) 3)
10 Angle Relationships Side Relationships
11 EX 1: Solve for x and y. ABC A x 10 cm ~ SLT y L 5 cm S B 24 cm C T 13 cm
12 EX 1: Solve for x and y. ABC A x 10 cm ~ SLT y L 5 cm S B 24 cm C T 13 cm
13 EX 2: ABCD ~ EFGH. Solve for x. D C H G A 6 x B E F
14 EX 2: ABCD ~ EFGH. Solve for x. D C H G A 6 x B E F
15 EX 3: A tree cast a shadow 18 feet long. At the same time a person who is 6 feet tall cast a shadow 4 feet long. How tall is the tree?
16 EX 3: A tree cast a shadow 18 feet long. At the same time a person who is 6 feet tall cast a shadow 4 feet long. How tall is the tree? tree's shadow tree's height person's shadow person's height
17 Setting Up the Ratio of Similar Polygons Corresponding Sides : Corresponding Sides Or Perimeter : Perimeter A : B 8/24/2018
18 Area : Area A 2 : B 2 Volume: Volume A 3 : B 3 8/24/2018
19 EX 4: The ratio of the perimeters of CAT to DOG is 3:2. Find the value of y. A 6 4 O y D G C 10 T REMEMBER! The ratio of the perimeters of two similar polygons equals the ratio of any pair of corresponding sides.
20 EX 4: The ratio of the perimeters of CAT to DOG is 3:2. Find the value of y. A 6 4 O y D G C 10 T
21 EX 5: Find the perimeter of the smaller triangle. 12 cm 4 cm Perimeter = 60 cm Perimeter = x
22 EX 5: Find the perimeter of the smaller triangle. 12 cm 4 cm Perimeter = 60 cm Perimeter = x
23 7.5 W Warm-Up Z 12 U 5 V X 6 Y 1. UVW ~ 2. What is the scale factor of UVW to XYZ 3. What is VW? 4. What is XZ? 5. If m U 50 and m Y 30, what is m Z?
24 7.5 W Warm-Up Z 12 U 5 1. UVW ~ XYZ What is VW? 4. What is XZ? V What is the scale factor of UVW to XYZ If m U 50 and m Y 30, what is m Z? 100 X 6 Y 5/6
25 Similarity Proving Triangles Similar 25
26 Shared angles are congruent in each triangle by the REFLEXIVE property. A A
27 Shared angles are congruent in each triangle by the REFLEXIVE property. Since A A, then, CAB DAE
28 VERTICAL ANGLES are the opposite angles created when two lines intersect one another. VERTICAL ANGLES ARE ALWAYS CONGRUENT
29 Before we start let s get a few things straight C Y A B X Z INCLUDED ANGLE It s stuck in between!
30 Two triangles are similar if all of their corresponding angles have measures and all of their corresponding sides have lengths. However, you do not need to know every one of those angle measures and side lengths to prove KEY IDEAS proportional equal that two triangles are similar. 30
31 KEY IDEAS (cont.) For instance, if two angles of one triangle are congruent similar to two angles of another triangle, then the triangles are. Also, if the three sides of one triangle are to the three sides of another triangle, then the triangles are. Finally, if two sides of one triangle are to two sides of another triangle and the angles of those sides are, then the similar triangles are. proportional congruent similar proportional included 31
32 Angle-Angle Similarity (AA~) Postulate If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
33 Side-Side-Side Similarity (SSS~) Theorem If the three sides of one triangle are proportional in length to the three sides of another triangle, then the triangles are similar.
34 Side-Angle-Side Similarity (SAS~) Theorem If two sides of one triangle have lengths that are proportional to two sides of another triangle and the included angles of those sides are congruent, then the triangles are similar.
35 Ex. Determine whether the triangles are similar. If so, tell which similarity test is used and complete the statement. G 43 M F H L 43 K V Y 7 3 W 11 U X 5 Z
36 Prove that RST~ PSQ 1. Two sides are proportional SAS~ S 2. Included angle is congruent 4 5 P Q S S reflexive R T
37 Using only the information given, can it be shown that CDE ~ FGE? A) Yes, by the AA~ Postulate B) Yes, by the SAS~ Theorem C) Yes, by the SSS~ Theorem D) No, not enough information is given. 37
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