Question

check whether ( 5,-2 ) , ( 6, 4) and ( 7, -2 ) are the vertices of an isosceles triangle
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Answer 1

Answer:

In an isosceles triangle, any 2 of the 3 sides are equal. Let the three points be P(5,-2), Q(6, 4) & R(7, -2)

In order to be isosceles,

Either

PQ = PR

or

PQ = QR

or

PR = QR

We calculate the value of PQ, QR & PR by distance formula

$Calculating \:PQ\\PQ = \sqrt ((x_{2} - x_{1}) ^ 2 + (y_{2} - y_{1}) ^ 2)$

$\sqrt (6- 5)² + (4 − (−2))²$

$\sqrt 1² + (6)²$

$\sqrt 1 +36$

$= \sqrt37$

Calculating QR

$QR =√(x₂-x₂)² + (1/₂ - y ₁)²$

$= \sqrt (7-6)² + (−2 − 4)²$

$= \sqrt1² + (-6)²$

$= \sqrt 1+36$

$= \sqrt 37$

$Calculating \: PR \\ PR= \sqrt(x₂-x₂)² + (y₂ −y ₁)² \\ = \sqrt (7-5)² + (−2 − (−2))² \\ = \sqrt2²+(-2+ 2)² \\ = \sqrt2² +0² \\ = \sqrt 2² \\ = 2$

Hence, PQ = √37, QR = √37, PR = 2

Since PQ =QR

It satisfies the condition of isosceles triangle

Hence, PQR is an isosceles triangle

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