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 ? (4 points)

Question

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 ? (4 points)
use numbers when explaining

GPT-4o mini · Answer

To determine if ΔABC and ΔEFG are similar, we first calculate the lengths of their sides. For ΔABC, we calculate the distances between the vertices: AB = √[(-3 + 4)² + (1 - 3)²] = √[1 + 4] = √5, BC = √[(-1 + 3)² + (2 - 1)²] = √[4 + 1] = √5, and CA = √[(-1 + 4)² + (2 - 3)²] = √[9 + 1] = √10.

Next, we calculate the lengths of the sides of ΔEFG: EF = √[(1.6 - 4.5)² + (3.8 - 5)²] = √[(-2.9)² + (-1.2)²] = √[8.41 + 1.44] = √9.85, FG = √[(3 - 1.6)² + (1.5 - 3.8)²] = √[(1.4)² + (-2.3)²] = √[1.96 + 5.29] = √7.25, and GE = √[(4.5 - 3)² + (5 - 1.5)²] = √[(1.5)² + (3.5)²] = √[2.25 + 12.25] = √14.5.

After comparing the ratios of the sides of both triangles, we find they are not proportional (ΔABC's sides are approximately 1:1:√2, while ΔEFG's sides do not maintain a consistent ratio), indicating that ΔEFG is not the image of ΔABC based on similarity criteria.