Answer these two questions in 3–5 sentences. Are the figures similar? ΔABC has vertices A(−4,3) , B(−3,1) , and C(−1,2) . ΔABC is rotated clockwise about the origin and then translated. ΔEFG has vertices E(4.5,5) , F(1.6,3.8) , and G(3,1.5) . Is ΔEFG the image of ΔABC?

Answer 1

To determine if the triangles ΔABC and ΔEFG are similar, we need to check if the ratios of their corresponding sides are equal. First, we calculate the lengths of the sides of ΔABC using the distance formula:

• Side AB: √[(-3 - (-4))^2 + (1 - 3)^2] = √[1^2 + (-2)^2] = √5
• Side BC: √[(-1 - (-3))^2 + (2 - 1)^2] = √[2^2 + 1^2] = √5
• Side CA: √[(-1 - (-4))^2 + (2 - 3)^2] = √[3^2 + (-1)^2] = √10

The ratios of ΔABC sides are not all equal. When determining the side lengths of ΔEFG using the same distance formula, we find that they yield different lengths compared to those of ΔABC. Since the corresponding sides differ in length, we conclude that ΔABC and ΔEFG are not similar and thus ΔEFG is not the image of ΔABC through rotation and translation.