Question

Triangle A B C is drawn with a solid line and is located in quadrant 2 of a coordinate plane. Triangle E F G is drawn with a dotted line and is located in quadrant 1. Angle A and Angle E are labeled as equivalent angles measuring 45 degrees.
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Vertex A is located at left parenthesis negative 4 comma 3 right parenthesis. Vertex B is located at left parenthesis negative 3 comma 1 right parenthesis, and vertex C is located at left parenthesis negative 1 comma 2 right parenthesis. Vertex E is located at approximately left parenthesis 4.5 comma 5 right parenthesis. Vertex F is located at approximately left parenthesis 1.5 comma 3.75 right parenthesis, and vertex G is located at approximately left parenthesis 3 comma 1.5 right parenthesis. The x-axis is labeled from negative 5 to 5 in increments of 1. The y-axis is labeled from 0 to 6 in increments of 1.


Answer the question in 3–5 sentences. Are the figures similar? ΔABC
has vertices A(−4,3)
, B(−3,1)
, and C(−1,2)
. ΔEFG
has vertices E(4.5, 5)
, F(1.5, 3.75)
, and G(3, 1.5)
.

Answer 1

No — the triangles are not similar. In ΔABC, AB = BC = √5 ≈ 2.236 and AC = √10 ≈ 3.162, so ABC is an isosceles 45°-45°-90° triangle. In ΔEFG the side lengths are EF ≈ 3.25, FG ≈ 2.704, and EG ≈ 3.808, so the two sides adjacent to angle E are not proportional to the two sides adjacent to angle A (EF/AB ≈ 1.455 while EG/AC ≈ 1.204). Because the corresponding side ratios are not equal (and EFG is not isosceles right), the triangles are not similar.