Given pqr is a right angled triangle, PQ= 16 cm, PR= 8 cm how do you calculate the length of qr?

Apr 25, 2018

We have either $Q {R}^{2} = P {Q}^{2} + P {R}^{2}$ giving $Q R = 8 \sqrt{5}$ or $P {Q}^{2} = Q {R}^{2} + P {R}^{2}$ giving $Q R = 8 \sqrt{3} .$

Explanation:

Let's follow the usual convention and call the triangle $P Q R$ with sides $p = Q R , q = P R , r = Q P$.

We're given $q = 8 , r = 16$ and $P Q R$ is a right triangle, so one of $P ,$ $Q ,$ or $R$ is ${90}^{\circ} .$

$q$ isn't the biggest side so can't be the hypotenuse. It's can be either $p$ or $r$ though. Let's work out both.

${p}^{2} = {q}^{2} + {r}^{2} = {8}^{2} + {16}^{2} = 5 \left({8}^{2}\right)$ so $p = 8 \sqrt{5}$

or

${r}^{2} = {p}^{2} + {q}^{2}$ so ${p}^{2} = {r}^{2} - {q}^{2} = {16}^{2} - {8}^{2} = 3 \left({8}^{2}\right)$ so $p = 8 \sqrt{3}$.