Question

consider the triangle shown. which shows the sides in order from longest to shortest? p 105° r 40° 35° q
o pq, rq, rp
o rq, pq, rp
o rq, rp, pq
o rp, pq, rq

Explanation:

Step1: Recall the triangle side - angle relationship

In a triangle, the larger the angle, the longer the side opposite to it. This is based on the principle that in a triangle, the side opposite a greater angle is longer.

Step2: Identify the angles and their opposite sides

• Angle at \( P \): \( \angle P = 105^{\circ} \), the side opposite to \( \angle P \) is \( RQ \).
• Angle at \( R \): \( \angle R = 40^{\circ} \), the side opposite to \( \angle R \) is \( PQ \).
• Angle at \( Q \): \( \angle Q = 35^{\circ} \), the side opposite to \( \angle Q \) is \( RP \).

Step3: Order the angles from largest to smallest

The angles in the triangle are \( 105^{\circ} (\angle P) \), \( 40^{\circ} (\angle R) \), and \( 35^{\circ} (\angle Q) \). So, the order of angles from largest to smallest is \( \angle P> \angle R > \angle Q \).

Step4: Order the sides based on the opposite angles

Since the side opposite a larger angle is longer, the order of the sides (from longest to shortest) will be the side opposite the largest angle, then the side opposite the next - largest angle, and then the side opposite the smallest angle.

• The side opposite \( \angle P(105^{\circ}) \) is \( RQ \).
• The side opposite \( \angle R(40^{\circ}) \) is \( PQ \).
• The side opposite \( \angle Q(35^{\circ}) \) is \( RP \).

So, the order of the sides from longest to shortest is \( RQ, PQ, RP \).

Answer:

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