7.2.3 Dilations and Similarity in the Coordinate Plane (continued) Example 2 Finding Coordinates of Similar Triangles The scale factor for a dilation of two similar
triangles in the coordinate plane is determined in this example. The unknown coordinates of a vertex are calculated. The coordinates of all but one of the vertices of the triangles are given. To determine the scale factor, set up a ratio of the lengths of two corresponding sides of the triangles. The length of OA (where O is the origin) is 2 and the length of OC is 6. The ratio of the sides, and the scale factor for the dilation, is 6/2, or 3. The coordinates for D are obtained by multiplying the scale factor by the coordinates for B . The coordinates for D are D (3 ⋅ (3), 0 ⋅ (3)) = D (9, 0). Two triangles in the coordinate plane are proved to be similar using the Side-Angle-Side Similarity Theorem in this example. The coordinates of the vertices of the triangles are given. Begin by recognizing that ∠ TPQ ≅ ∠ SPR , by the Reflexive Property (they are the same angle). Calculate the lengths of PQ and PR by subtracting the y coordinates of the endpoints: PQ = 3 and PR = 6. The similarity ratio of the side lengths is 6/3 = 2. Using the Distance Formula, calculate the lengths of PT and PS . The similarity ratio of these side lengths is also 2. Since two pairs of corresponding sides are similar and the included angle is congruent, △ PQT ∼ △ PRS by the Side-Angle-Side Similarity Theorem.
Example 3 Proving Triangles are Similar
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