Prove the A \left(\right. - 5 , 4 \left.\right) , B \left(\right. - 1 , - 2 \left.\right) , C \left(\right. 5 , 2 \left.\right) are the vertices of n isosceles right angled triangle ?

Engineering

Mathematics

Section Formulae and Centres of a Triangle

Various Form of a Straight Line

Length of Perpendicular

Question

Prove the

A (- 5, 4), B (- 1, - 2), C (5, 2)

are the vertices of n isosceles right angled triangle ?

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A (- 5, 4), B (- 1, - 2), C (5, 2) \vec{A B} = 4 \hat{i} - 6 \hat{j}, \vec{B C} = 6 \hat{i} + 4 \hat{j}, \vec{C A} = 10 \hat{i} - 2 \hat{j} ∣ \vec{A B}, ∣ = \sqrt{4^{2} + 6^{2}} = 2 \sqrt{13} ∣ \vec{B C}, ∣ = \sqrt{6^{2} + 4^{5}} = 2 \sqrt{13} ∣ \vec{C A}, ∣ = \sqrt{10^{2} + 2^{2}} = 2 \sqrt{26}

∣ \vec{A B}, ∣ = ∣ \vec{B C}, ∣ ∴ ∠ A = ∠ C

∴

isoscles triangle.

{∣ \vec{A B}, ∣,}^{2} + {∣ \vec{B C}, ∣,}^{2} = {∣ \vec{C A}, ∣,}^{2} \Rightarrow

right angled at

B

∴

The triangle is right angles isoscles triangle.