Question

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. Triangle ABC is isosceles while triangle EFG is not, so they cannot be similar.

Compute side lengths:
- AB = distance A(−4,3) to B(−3,1) = sqrt(1^2+(-2)^2) = sqrt5.
- BC = distance B(−3,1) to C(−1,2) = sqrt(2^2+1^2) = sqrt5.
- AC = sqrt(3^2+(-1)^2) = sqrt10.

So AB = BC (isosceles).

For EFG:
- EF = distance (4.5,5) to (1.5,3.75) = sqrt(10.5625) = 3.25.
- FG = distance (1.5,3.75) to (3,1.5) = sqrt(7.3125) ≈ 2.704.
- EG = distance (4.5,5) to (3,1.5) = sqrt(14.5) ≈ 3.808.

No two sides in EFG are equal, so EFG is scalene. Because ABC has two equal sides but EFG does not, the triangles are not similar.