Given a triangle with angle coordinates: A(0,0), B(3,4), C(7,1) Prove that triangleABC is isosceles.

Given a triangle with angle coordinates: A(0,0), B(3,4), C(7,1) Prove that triangleABC is isosceles.

Question
Given a triangle with angle coordinates:
A(0,0), B(3,4), C(7,1)
Prove that \(\displaystyle\triangle{A}{B}{C}\) is isosceles.

Answers (1)

2021-02-11
An isosceles triangle has at least two sides of equal length. Let's determine the length of each side using the distance formula:
\(\displaystyle{d}=\sqrt{{{\left({\underset{{{2}}}{{{x}}}}-{\underset{{{1}}}{{{x}}}}\right)}^{{2}}+{\left({\underset{{{2}}}{{{y}}}}-{\underset{{{1}}}{{{y}}}}\right)}^{{2}}}}\)
Substitute for AB:
\(\displaystyle{A}{B}=\sqrt{{{\left({3}-{0}\right)}^{{2}}+{\left({4}-{0}\right)}^{{2}}}}=\sqrt{{{9}+{16}}}=\sqrt{{25}}={5}\)
Substitute for BC:
\(\displaystyle{B}{C}=\sqrt{{{\left({7}-{3}\right)}^{{2}}+{\left({1}-{4}\right)}^{{2}}}}=\sqrt{{{16}+{9}}}=\sqrt{{25}}={5}\)
Since \(\displaystyle{A}{B}={B}{C},\triangle{A}{B}{C}\) is isosceles.
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