Question

The vertices of ΔABC are A(1,–2), B(1,1), and C(5,–2). Which could be the side lengths of a triangle that is similar but not congruent to ΔABC?

3, 4, and 5 units
9, 12, and 15 units
6, 4, and 5 units
9, 12, and 5 units

Answer 1

Compute the side lengths of ΔABC:
AB = distance between (1,−2) and (1,1) = 3,
AC = distance between (1,−2) and (5,−2) = 4,
BC = distance between (1,1) and (5,−2) = 5.

So ΔABC is a 3–4–5 triangle. A similar but not congruent triangle must have side lengths proportional to 3,4,5 with scale factor ≠ 1. Only 9, 12, 15 = 3·(3,4,5) fits.