Question

three friends arrange to meet at a park entrance, but each friend ends up at a different entrance, as shown in the diagram. pat is at a right angle from chris and jaime, and chris is at a 48° angle from pat and jaime. which statements are true? check all that apply. jaime is at the smallest angle from the two friends. pat is at an angle of 90° to chris and jaime. the path between jaime and chris is the shortest because it is adjacent to the smallest angles. if they meet at pats entrance, then chris will have walked a shorter distance than jaime. if they meet at jaimes entrance, then pat will have walked a longer distance than chris.

Explanation:

Step1: Find the third angle

In a right - triangle (Pat, Chris, Jaime), the sum of angles is \(180^{\circ}\). Given one angle is \(90^{\circ}\) (Pat's angle) and another is \(48^{\circ}\) (Chris's angle). Let the angle at Jaime be \(x\). Then \(x = 180-(90 + 48)=42^{\circ}\).

Step2: Analyze each statement

• Jaime is at the smallest angle from the two friends:

Since \(42^{\circ}<48^{\circ}<90^{\circ}\), the angle at Jaime (\(42^{\circ}\)) is the smallest. This statement is true.

• Pat is at an angle of \(90^{\circ}\) to Chris and Jaime:

The problem states "Pat is at a right angle from Chris and Jaime", so this statement is true.

• The path between Jaime and Chris is the shortest because it is adjacent to the smallest angles:

In a triangle, the side opposite the smallest angle is the shortest. The side opposite the \(42^{\circ}\) (Jaime's angle) is the side between Pat and Chris. So this statement is false.

• If they meet at Pat's entrance, then Chris will have walked a shorter distance than Jaime:

In a right - triangle, using the fact that in a right - triangle with angles \(42^{\circ},48^{\circ},90^{\circ}\), the side opposite \(48^{\circ}\) (Chris - Pat - Jaime triangle, side from Chris to Pat) and side opposite \(42^{\circ}\) (Jaime to Pat). Since \(\sin42=\frac{\text{opposite (Chris - Pat)}}{\text{hypotenuse (Jaime - Chris)}}\) and \(\sin48=\frac{\text{opposite (Jaime - Pat)}}{\text{hypotenuse (Jaime - Chris)}}\), and \(\sin42<\sin48\), the side from Chris to Pat is shorter than the side from Jaime to Pat. This statement is true.

• If they meet at Jaime's entrance, then Pat will have walked a longer distance than Chris:

The side from Pat to Jaime (opposite \(48^{\circ}\)) and the side from Chris to Jaime (hypotenuse). Also, using \(\cos48=\frac{\text{adjacent (Chris - Pat)}}{\text{hypotenuse (Chris - Jaime)}}\), the side from Pat to Jaime (\(\text{opposite of }48^{\circ}\)) is longer than the side from Chris to Pat (\(\text{adjacent of }48^{\circ}\)). Since the side from Chris to Jaime (hypotenuse) is the longest side. But comparing Pat - Jaime and Chris - Jaime: \(\text{Pat - Jaime}=\text{hypotenuse}\times\sin48\) and \(\text{Chris - Jaime}=\text{hypotenuse}\). So Pat (Pat - Jaime) has walked a shorter distance than Chris (Chris - Jaime). This statement is false.

Answer:

Jaime is at the smallest angle from the two friends, Pat is at an angle of \(90^{\circ}\) to Chris and Jaime, If they meet at Pat's entrance, then Chris will have walked a shorter distance than Jaime.