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:

1. Jaime's angle: In a triangle, the sum of angles is \(180^\circ\). The triangle here has a right angle (\(90^\circ\)) at Pat and \(48^\circ\) at Chris, so Jaime's angle is \(180 - 90 - 48 = 42^\circ\), which is the smallest. So this statement is true.
2. Pat's angle: The diagram shows a right angle at Pat, so Pat is at \(90^\circ\) to Chris and Jaime. This statement is true.
3. Shortest path (Jaime - Chris): In a triangle, the shortest side is opposite the smallest angle. The smallest angle is \(42^\circ\) (at Jaime), so the side opposite (Pat - Chris) is the shortest, not Jaime - Chris. So this statement is false.
4. Meeting at Pat's entrance: The side opposite the smallest angle (\(42^\circ\) at Jaime) is Pat - Chris, and the side opposite \(48^\circ\) (at Chris) is Pat - Jaime. Since \(42^\circ<48^\circ\), Pat - Chris is shorter than Pat - Jaime. So Chris walks shorter than Jaime. This statement is true.
5. Meeting at Jaime's entrance: The side opposite \(90^\circ\) (at Pat) is Jaime - Chris, and the side opposite \(48^\circ\) (at Chris) is Jaime - Pat. Since \(48^\circ < 90^\circ\), Jaime - Pat is shorter than Jaime - Chris. So Pat walks shorter than Chris, not longer. 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.