Question

applying theorems to a real-world problem
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? choose three correct answers.
jaime is at the smallest angle from the two friends.
if they meet at jaimes entrance, then pat will have walked a longer distance than chris.
if they meet at pats entrance, then chris will have walked a shorter distance than jaime.
pat is at an angle of 90° to chris and jaime.

Explanation:

• For the first statement: In a right - triangle (Pat, Chris, Jaime), the angles are \(90^{\circ}\), \(48^{\circ}\), and \(180-(90 + 48)=42^{\circ}\). So Jaime's angle (\(42^{\circ}\)) is the smallest.
• For the second statement: In a right - triangle, the side opposite the larger non - right angle is longer. When meeting at Jaime's entrance, using the property that in a right - triangle \(\triangle Pat - Chris - Jaime\) with \(\angle Pat = 90^{\circ}\), \(\angle Chris=48^{\circ}\), \(\angle Jaime = 42^{\circ}\). The side opposite \(\angle Chris\) (Pat's path) is longer than the side opposite \(\angle Jaime\) (Chris's path) by the sine rule (\(\frac{\text{Pat - Jaime}}{\sin48^{\circ}}=\frac{\text{Chris - Jaime}}{\sin42^{\circ}}\), and \(\sin48^{\circ}>\sin42^{\circ}\)).
• For the fourth statement: Given in the problem, Pat is at a right - angle (\(90^{\circ}\)) from Chris and Jaime.

Answer:

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