Question

semester 1
consider the triangle shown.
triangle image with angles at p: 105°, at r: 40°, at q: 35°
which shows the sides in order from longest to shortest?
options:

• $overline{rp}$, $overline{pq}$, $overline{rq}$
• $overline{pq}$, $overline{rq}$, $overline{rp}$
• $overline{rq}$, $overline{pq}$, $overline{rp}$
• $overline{rq}$, $overline{rp}$, $overline{pq}$

Explanation:

Step1: Find the third angle

In a triangle, the sum of angles is \(180^\circ\). Given angles at \(P = 105^\circ\), at \(R = 40^\circ\), so angle at \(Q\) is \(180 - 105 - 40 = 35^\circ\).

Step2: Relate angles to sides

In a triangle, the larger the angle, the longer the opposite side. Angles: \( \angle P = 105^\circ\) (largest), \( \angle R = 40^\circ\), \( \angle Q = 35^\circ\) (smallest). Opposite sides: \(RQ\) (opposite \( \angle P\)), \(PQ\) (opposite \( \angle R\)), \(RP\) (opposite \( \angle Q\)). So side lengths from longest to shortest: \(RQ\), \(PQ\), \(RP\).

Answer:

\(\overline{RQ}, \overline{PQ}, \overline{RP}\) (the option with this order)