Decide whether or not the points are the vertices of a right triangle.Consider the three points A = (-5, -2), B = (-7, 0), C = (-11, -4). Determine whether the triangle ABC is a right triangle. Explain your reasoning.
A. The side lengths of triangle ABC are d(A, B) = 4
, d(A, C) = 2
, d(B, C) = 4.
[d(A, B)]2 + [d(B, C)]2 = (4
)2 + 42 = 32 + 16 = 48
[d(A, C)]2 = (2
)2 = 40
Since [d(A, C)]2 ? [d(A, B)]2 + [d(B, C)]2, triangle ABC is not a right triangle.
B. The side lengths of triangle ABC are d(A, B) = 2
, d(A, C) = 2
, d(B, C) = 4.
[d(A, B)]2 + [d(B, C)]2 = (2
)2 + 42 = 8 + 16 = 24
[d(A, C)]2 = (2
)2 = 40
Since [d(A, C)]2 ? [d(A, B)]2 + [d(B, C)]2 , triangle ABC is not a right triangle.
C. The side lengths of triangle ABC are d(A, B) = 2
, d(A, C) = 4
, d(B, C) = 4
.
[d(A, B)]2 + [d(B, C)]2 = (2
)2 + (4
)2 = 8 + 32 = 40
[d(A, C)]2 = (4
)2 = 160
Since [d(A, C)]2 ? [d(A, B)]2 + [d(B, C)]2, triangle ABC is not a right triangle.
D. The side lengths of triangle ABC are d(A, B) = 2
, d(A, C) = 2
, d(B, C) = 4
.
[d(A, B)]2 + [d(B, C)]2 = (2
)2 + (4
)2 = 8 + 32 = 40
[d(A, C)]2 = (2
)2 = 40
Since [d(A, C)]2 = [d(A, B)]2 + [d(B, C)]2, triangle ABC is a right triangle.
Answer: D
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