Question

In a $△PQR,$ perpendicular $PS$ from $P$ to $QR$ meets $QR$ at $S.$ If $PS:QS:RS=2:4:1,$ then which of the following is correct$?$

• A. $PQR$ is an equilateral triangle
• B. $PQR$ is right angled at $P$
• C. $PQR$ is an isosceles triangle
• D. $PQ=3PR$

Answer 1

Answer: (B)

$PQR$ is right angled at $P$

Given, $PS\perp QR$
Let $PS=2x,QS=4x,RS=x$
Now, in $△PSR$
$P{R}^2=P{S}^2+R{S}^2$
$P{R}^2=4x^2+x^2$
$PR=\sqrt{5}x$
In $△PQS$
$P{Q}^2=P{S}^2+Q{S}^2$
$P{Q}^2=4x^2+16x^2$
$PQ=\sqrt{20}x$
Now, $P{Q}^2+P{R}^2=20x^2+5x^2=25x^2$
and, $P{S}^2=(4x+x{)}^2=25x^2$
Thus, $P{Q}^2+P{R}^2=P{S}^2$
By converse of Pythagoras theorem, the triangle is a right angled triangle.