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(1.6,1.8), and G(3,1.5) Is ΔEFG the image of ΔABC?
To determine if ΔEFG is the image of ΔABC, we need to check if the figures have the same shape and size.
First, let's compare the shape. To do this, we can calculate the slopes of the sides of both triangles. The slope of BC can be calculated as (1-3)/(-3-(-4)) = 2/-1 = -2, and the slope of FG can be calculated as (1.8-1.5)/(1.6-3) = 0.3/-1.4 ≈ -0.214. Since the slopes are not equal, the sides are not parallel, and therefore the shape is not the same.
Next, we can compare the size. We can use the distance formula to calculate the lengths of the sides.
The length of BC can be calculated as √((-3-(-4))^2 + (1-3)^2) = √((-3+4)^2 + (-2)^2) = √(1+4) = √5 ≈ 2.236
The length of FG can be calculated as √((1.6-3)^2 + (1.8-1.5)^2) = √((-1.4)^2 + 0.3^2) = √(1.96+0.09) = √2.05 ≈ 1.43
Since the lengths of the sides are different, the sizes are not the same.
Therefore, ΔEFG is not the image of ΔABC.