Question

the lacrosse player shown passes the ball to a team member who is directly in front of the center of the net and is directly below her in the diagram the total distance of the two throws? the corners of the net are at (15, 27) and (15, 33). enter the correct value in the box.

Explanation:

Step1: Find the center of the net

The net's corners are at \((15, 27)\) and \((15, 33)\). The center has the same \(x\)-coordinate (15) and the \(y\)-coordinate is the average of 27 and 33. So, \(y=\frac{27 + 33}{2}=\frac{60}{2}=30\). The center is \((15, 30)\).

Step2: Find the first throw distance (from \((45, 50)\) to \((15, 30)\))

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here, \(x_1 = 45,y_1 = 50,x_2 = 15,y_2 = 30\). So, \(d_1=\sqrt{(15 - 45)^2+(30 - 50)^2}=\sqrt{(- 30)^2+(-20)^2}=\sqrt{900 + 400}=\sqrt{1300}=10\sqrt{13}\approx36.06\)? Wait, no, wait. Wait, the team member is directly below the lacrosse player? Wait, no, the lacrosse player is at \((45,50)\), and the team member is directly in front of the center of the net and directly below her. Wait, maybe the team member is at \((45, 30)\)? Wait, no, the center of the net is at \((15,30)\), but the team member is directly in front of the center of the net and directly below the lacrosse player (who is at \((45,50)\)). So the team member has the same \(x\)-coordinate as the lacrosse player? Wait, no, "directly in front of the center of the net" – maybe the center of the net is at \((15,30)\), and the team member is on the same vertical line as the center? No, the problem says "directly in front of the center of the net and is directly below her in the diagram". So the lacrosse player is at \((45,50)\), the team member is directly below her, so same \(x\)-coordinate (45), and directly in front of the center of the net. The center of the net is at \((15,30)\), so "in front of" might mean same \(y\)-coordinate as the center? Wait, no, let's re - read.

Wait, the corners of the net are at \((15,27)\) and \((15,33)\), so the net is vertical (same \(x = 15\)), from \(y = 27\) to \(y = 33\). The center of the net is at \((15,30)\) (midpoint). The team member is directly in front of the center of the net (so same \(y\)-coordinate as center, 30) and directly below the lacrosse player (who is at \((45,50)\)), so same \(x\)-coordinate as lacrosse player, 45? Wait, no, "directly below" would mean same \(x\)-coordinate, and "directly in front of the center of the net" – maybe "in front of" is along the \(y\)-axis. Wait, maybe the first throw is from \((45,50)\) to \((45,30)\) (directly below, so vertical distance: \(50 - 30=20\) yards), and the second throw is from \((45,30)\) to \((15,30)\) (horizontal distance: \(45 - 15 = 30\) yards). Wait, that makes more sense. Because the center of the net is at \((15,30)\), and the team member is directly below the lacrosse player (so \(x = 45\), \(y = 30\)), then from team member to center of net is horizontal.

Wait, let's check the coordinates again. The lacrosse player is at \((45,50)\). The team member is directly below her, so \(x = 45\), and directly in front of the center of the net. The center of the net is at \((15,30)\), so "in front of" the center would be same \(y\)-coordinate (30), because the net is at \(x = 15\), so "in front of" the net (towards the lacrosse player) would be increasing \(x\)-coordinate, but the team member is directly below the lacrosse player, so \(x = 45\), \(y = 30\). Then the first throw is from \((45,50)\) to \((45,30)\): distance is \(|50 - 30|=20\) yards. The second throw is from \((45,30)\) to \((15,30)\): distance is \(|45 - 15| = 30\) yards. Then total distance is \(20+30 = 50\) yards. Wait, that seems too simple. Wait, maybe I misinterpret the coordinates.

Wait, another approach: The center of the net: midpoint of \((15,27)\) and \((15,33)\) is \((15,\frac{27 + 33}{2})=(15,30)\). The lacrosse player is at \((45,50)\). The team member is directly in front of the center of the net (so same \(y\)-coordinate as center, 30) and directly below the lacrosse player (same \(x\)-coordinate as lacrosse player, 45). So team member is at \((45,30)\).

First throw: from \((45,50)\) to \((45,30)\). Using distance formula: \(d_1=\sqrt{(45 - 45)^2+(30 - 50)^2}=\sqrt{0+(-20)^2}=20\) yards.

Second throw: from \((45,30)\) to \((15,30)\). Using distance formula: \(d_2=\sqrt{(15 - 45)^2+(30 - 30)^2}=\sqrt{(-30)^2+0}=30\) yards.

Total distance: \(d_1 + d_2=20 + 30=50\) yards.

Answer:

50