Question

consider the triangle shown. which shows the sides in order from longest to shortest?
triangle with vertices p, r, q; angles at p: 105°, at r: 40°, at q: 35°
options:
$overline{pq}$, $overline{rq}$, $overline{rp}$
$overline{rq}$, $overline{pq}$, $overline{rp}$
$overline{rq}$, $overline{rp}$, $overline{pq}$
$overline{rp}$, $overline{pq}$, $overline{rq}$

Explanation:

Step1: Recall the triangle side - angle relationship

In a triangle, the larger the angle opposite a side, the longer the side. So we first need to identify the angles opposite each side.

• In $\triangle PQR$, angle at $P$ is $\angle P = 105^{\circ}$, angle at $R$ is $\angle R=40^{\circ}$, angle at $Q$ is $\angle Q = 35^{\circ}$.
• Side opposite $\angle P$ is $\overline{RQ}$.
• Side opposite $\angle Q$ is $\overline{RP}$.
• Side opposite $\angle R$ is $\overline{PQ}$.

Step2: Order the angles from largest to smallest

We compare the measures of the angles: $105^{\circ}>40^{\circ}>35^{\circ}$, so $\angle P>\angle R>\angle Q$.

Step3: Order the sides based on the angle - side relationship

Since the side opposite a larger angle is longer, the side opposite $\angle P$ ($\overline{RQ}$) is longer than the side opposite $\angle R$ ($\overline{PQ}$), and the side opposite $\angle R$ ($\overline{PQ}$) is longer than the side opposite $\angle Q$ ($\overline{RP}$). So the order of the sides from longest to shortest is $\overline{RQ},\overline{PQ},\overline{RP}$.

Answer:

$\boldsymbol{\overline{RQ},\overline{PQ},\overline{RP}}$ (corresponding to the option $\boldsymbol{\overline{RQ},\overline{PQ},\overline{RP}}$)