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? choose three correct answers. if they meet at pat’s entrance, then chris will have walked a shorter distance than jaime. the path between jaime and chris is the shortest because it is adjacent to the smallest angles. 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° to chris and jaime

Explanation:

Step1: Analyze the triangle

The diagram forms a right - triangle with Pat, Chris, and Jaime. Let's denote the right - angle at Pat. So, triangle \( \triangle \text{Pat - Chris - Jaime} \) is a right - triangle with \( \angle \text{Pat - Chris - Jaime}=48^{\circ} \), \( \angle \text{Pat}=90^{\circ} \), and we can find \( \angle \text{Jaime} = 180^{\circ}-90^{\circ}-48^{\circ}=42^{\circ} \).

Step2: Analyze the first statement

Statement 1: "If they meet at Pat’s entrance, then Chris will have walked a shorter distance than Jaime."
In a right - triangle, the side opposite the smaller angle is shorter. \( \angle \text{Jaime}=42^{\circ} \), \( \angle \text{Chris}=48^{\circ} \). The side opposite \( \angle \text{Jaime} \) is \( \text{Chris - Pat} \), and the side opposite \( \angle \text{Chris} \) is \( \text{Jaime - Pat} \). Since \( 42^{\circ}<48^{\circ} \), \( \text{Chris - Pat}<\text{Jaime - Pat} \). So if they meet at Pat's entrance, Chris walks \( \text{Chris - Pat} \) and Jaime walks \( \text{Jaime - Pat} \), so Chris walks a shorter distance. This statement is correct.

Step3: Analyze the second statement

Statement 2: "The path between Jaime and Chris is the shortest path because it is adjacent to the smallest angles."
The shortest path between two points is a straight line. In a triangle, the hypotenuse (the side between Jaime and Chris) is the longest side, not the shortest. Also, the angles adjacent to the hypotenuse are the non - right angles (\( 42^{\circ} \) and \( 48^{\circ} \)), but the hypotenuse is the longest side. So this statement is incorrect.

Step4: Analyze the third statement

Statement 3: "Jaime is at the smallest angle from the two friends."
We found \( \angle \text{Jaime}=42^{\circ} \), \( \angle \text{Chris}=48^{\circ} \), \( \angle \text{Pat}=90^{\circ} \). Since \( 42^{\circ} \) is the smallest angle among \( 42^{\circ} \), \( 48^{\circ} \), and \( 90^{\circ} \), Jaime is at the smallest angle (the angle at Jaime) from the two friends (Pat and Chris). This statement is correct.

Step5: Analyze the fourth statement

Statement 4: "If they meet at Jaime’s entrance, then Pat will have walked a longer distance than Chris."
In the right - triangle, the side opposite \( \angle \text{Chris}=48^{\circ} \) is \( \text{Jaime - Pat} \), and the side opposite \( \angle \text{Jaime}=42^{\circ} \) is \( \text{Chris - Pat} \). Since \( 48^{\circ}>42^{\circ} \), \( \text{Jaime - Pat}>\text{Chris - Pat} \). If they meet at Jaime's entrance, Pat walks \( \text{Pat - Jaime} \) and Chris walks \( \text{Chris - Jaime} \)? No, wait, if they meet at Jaime's entrance, Pat walks \( \text{Pat - Jaime} \) and Chris walks \( \text{Chris - Jaime} \)? No, in the right - triangle, \( \text{Pat - Jaime} \) is opposite \( \angle \text{Chris} \), \( \text{Chris - Jaime} \) is the hypotenuse. Wait, no, when meeting at Jaime's entrance, Pat's path is \( \text{Pat - Jaime} \), Chris's path is \( \text{Chris - Jaime} \). We know that \( \text{Pat - Jaime} \) (opposite \( 48^{\circ} \)) and \( \text{Chris - Pat} \) (opposite \( 42^{\circ} \)). Also, by Pythagoras, \( \text{Chris - Jaime}=\sqrt{(\text{Pat - Chris})^2+(\text{Pat - Jaime})^2} \). But we know that \( \text{Pat - Jaime}>\text{Pat - Chris} \) (since \( 48^{\circ}>42^{\circ} \)). Wait, no, let's re - evaluate. The angle at Chris is \( 48^{\circ} \), so \( \sin(48^{\circ})=\frac{\text{Pat - Jaime}}{\text{Chris - Jaime}} \), \( \cos(48^{\circ})=\frac{\text{Pat - Chris}}{\text{Chris - Jaime}} \). Since \( \sin(48^{\circ})>\cos(48^{\circ}) \) (because \( 48^{\circ}>45^{\circ} \) and \( \sin\theta>\cos\theta \) for \( \theta > 45^{\circ} \) in \( 0^{\circ}<\theta<90^{\circ} \)), \( \text{Pat - Jaime}>\text{Pat - Chris} \). If they meet at Jaime's entrance, Pat walks \( \text{Pat - Jaime} \), Chris walks \( \text{Chris - Jaime} \)? No, no, the original positions: Pat, Chris, Jaime form a right - triangle with right - angle at Pat. So Pat is at one vertex, Chris at another, Jaime at the third. So the paths: Pat to Chris, Pat to Jaime, and Chris to Jaime (hypotenuse). If they meet at Jaime's entrance, Pat walks from Pat to Jaime, Chris walks from Chris to Jaime. Since \( \text{Pat - Jaime} \) is a leg, \( \text{Chris - Jaime} \) is the hypotenuse, so \( \text{Pat - Jaime}<\text{Chris - Jaime} \). Wait, this is the opposite of what the statement says. So this statement is incorrect.

Step6: Analyze the fifth statement

Statement 5: "Pat is at an angle of \( 90^{\circ} \) to Chris and Jaime."
Since the triangle has a right - angle at Pat, \( \angle \text{Pat}=90^{\circ} \), so Pat is at a \( 90^{\circ} \) angle to both Chris and Jaime (the sides \( \text{Pat - Chris} \) and \( \text{Pat - Jaime} \) are perpendicular). This statement is correct.

Answer:

The correct statements are:

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