ANSWER 3 2 1. Given RST XYZ with ~ RS XY ST YZ 3 2 RT XZ = = . Find . 2. Given EFG MNP with ~ ANSWER 2.25 EF MN 2 1 = . If FG = 4.5, find NP.

Show that triangles are similar. Target Show that triangles are similar. You will… Identify similarity transformations called dilations. .

Vocabulary dilation – a transformation that preserves angle measures and proportional side lengths; sometimes called a similarity transformation scale factor of dilation – ratio of side lengths image : preimage descriptions also include a “center” (point of dilation) dilations and similarity – if a dilation can be used to move one figure onto another, the two figures are similar. dilations and rigid motions – if a dilation followed by any combination of rigid motions can be used to move one figure onto the other, the two figures are similar.

EXAMPLE 1 Describe a dilation FEG is similar to FDH. Describe the dilation that moves FEG onto FDH. preimage image SOLUTION The figure shows a dilation with center F. The scale factor (image : preimage) is 𝟐 𝟏 because the ratio of FH : FG is 20 : 10 or 2 : 1.

EXAMPLE 2 Describe a combination of transformations ABC is similar to FGE. Describe a combination of transformations that moves ABC onto FGE. In the figure: AB = 9, FG = 6 image reduce (dilation) preimage SOLUTION A dilation with center B and scale factor moves ABC onto DBE 2 3 image:preimage Then a rotation of DBE with center E moves DBE onto FGE. The angle of rotation is equal to the measure of C.

GUIDED PRACTICE for Examples 1 and 2 The two figures are similar. Describe the transformation(s) that move the blue figure onto the red figure. 1. 2. ANSWER ANSWER dilation with center B and scale factor dilation with center D, scale factor and reflection 7 3 2 3

EXAMPLE 3 Use transformations to show figures are not similar Use transformations to explain why ABCDE and KLQRP are not similar. (The red polygon is the image.) The scale factor of the dilation with center A is 2 : 3. However, the angle measures have not been preserved with angles P, Q and R.

EXAMPLE 3 Use transformations to show figures are not similar SOLUTION Corresponding sides in the pentagons are proportional with a scale factor of . 2 3 However, this does not necessarily mean the pentagons are similar. A dilation with center A and scale factor moves ABCDE onto AFGHJ. Then a reflection moves AFGHJ onto KLMNP. 2 3 KLMNP does not exactly coincide with KLQRP, because not all of the corresponding angles are congruent. (Only A and K are congruent.) Since angle measure is not preserved, the two pentagons are not similar.

EXAMPLE 4 Use similar figures GRAPHIC DESIGN: a design for a party mask is made using all equilateral triangles and a scale factor of . 1 2 a. Describe transformations that move triangle A onto triangle B. b. Describe why triangles C and D are similar by using the given information.

EXAMPLE 4 Use similar figures Describe the transformations that move triangle A onto triangle B. SOLUTION The figure shows a dilation with scale factor . followed by a clockwise rotation of 60°. 1 2

EXAMPLE 4 Use similar figures Describe why triangles C and D are similar by using the given information. SOLUTION b. Triangles C and D are similar because all pairs of corresponding sides are proportional with a ratio of and all pairs of corresponding angles of equilateral triangles have the same measure. 1 2

GUIDED PRACTICE for Examples 3 and 4 Refer to the floor tile designs shown below. In each design, the red shape is a regular hexagon. Tile design 1 is made using two hexagons. Explain why the red and blue hexagons are not similar. The red hexagon has all sides congruent, but the blue hexagon has 3 shorter sides and 3 longer sides, so ratios of corresponding side lengths are not constant.

GUIDED PRACTICE for Examples 3 and 4 Tile design 2 is made using two similar geometric shapes. Describe the transformations that move the blue hexagon to the red hexagon. dilation followed by a rotation of 30° about the center of the figures

GUIDED PRACTICE for Examples 3 and 4 Tile design 3 shows congruent angles and sides. Explain why the red and blue hexagons are similar, using the given information. All angles are congruent, so angle measure is preserved, and all side lengths are congruent in each hexagon, so the ratio of any two corresponding side lengths is constant.

GUIDED PRACTICE for Examples 3 and 4 If the lengths of all the sides of one polygon are proportional to the lengths of all the corresponding sides of another polygon, must the polygons be similar? Explain. No; even though corresponding sides might be proportional, if corresponding angles are not congruent, the polygons are not similar.