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) 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?
To determine if the figures are similar, we need to compare the side lengths and the angles of the two triangles. To rotate ΔABC clockwise about the origin, we use the rotation matrix:
[cosθ -sinθ]
[sinθ cosθ]
Let's assume θ = 90 degrees. Applying the rotation matrix to each vertex of ΔABC, we get the image of ΔABC after rotation:
A' = (3, 4)
B' = (1, 3)
C' = (2, 1)
Next, we need to translate the image of ΔABC. The translation vector is determined by subtracting corresponding coordinates of a vertex in the original triangle from the corresponding coordinates of the corresponding vertex in the image after rotation. In this case, the translation vector is:
T = (4.5 - 3, 5 - 4) = (1.5, 1)
Adding the translation vector to each vertex of the image after rotation, we obtain the final image of ΔABC after both rotation and translation:
A'' = (4.5, 5)
B' = (2.5, 4)
C' = (3.5, 2)
Comparing the vertices of ΔEFG and the final image of ΔABC, we can see that they are not the same, so ΔEFG is not the image of ΔABC. Hence, the figures are not similar.