Question

In an equilateral triangle PQR, PT is an altitude. Then the value of 4PT² is:
(a) 3PQ ² (b) PQ² (c) (PQ + QR)² (d) 2PQ²​

Answer 1

$\underline{\textbf{Given:}}$

$\textsf{In equilateral triangle PQR, PT is an altitude}$

$\underline{\textbf{To find:}}$

$\textsf{The value of}\;\mathsf{4\,PT^2}$

$\underline{\textbf{Solution:}}$

$\underline{\textbf{Formula used:}}$

$\textsf{Altitude of equilateral triangle having side a is}$

$\mathsf{\dfrac{\sqrt3}{2}\,a}$

$\mathsf{Here,}$

$\mathsf{Altitude,\;PT=\dfrac{\sqrt3}{2}\,a}$

$\implies\mathsf{PT=\dfrac{\sqrt3}{2}\,PQ}$

$\textsf{Squaring on bothsides, we get}$

$\mathsf{PT^2=\left(\dfrac{\sqrt3}{2}\right)^2PQ^2}$

$\mathsf{PT^2=\dfrac{3}{4}PQ^2}$

$\implies\boxed{\mathsf{4\;PT^2=3\;PQ^2}}$

$\underline{\textbf{Answer:}}$

$\mathsf{Option\;(a)\;is\;correct}$