Question

1. δpqr is a right triangle. if pq = 8, what is pr? options: ( 8sqrt{3} ), 16, 4, ( 4sqrt{3} )

Explanation:

Step1: Identify triangle type and angles

$\triangle PQR$ is right - angled at $R$. $\angle P = 30^{\circ}$, $\angle Q=60^{\circ}$, and hypotenuse $PQ = 8$. We know that in a $30 - 60 - 90$ right - triangle, the sides are in the ratio $1:\sqrt{3}:2$, where the side opposite $30^{\circ}$ is the shortest side (let's call it $x$), the side opposite $60^{\circ}$ (which is $PR$ in this case) is $x\sqrt{3}$, and the hypotenuse is $2x$.

Step2: Relate hypotenuse to the ratio

The hypotenuse $PQ = 8$, and from the ratio of $30 - 60 - 90$ triangle, hypotenuse $= 2x$. So, $2x=8$. Solving for $x$, we divide both sides by 2: $x = 4$.

Step3: Find the length of $PR$

The side $PR$ is opposite the $60^{\circ}$ angle, so its length is $x\sqrt{3}$. Since $x = 4$, then $PR=4\sqrt{3}$.

Answer:

$4\sqrt{3}$