Question

ms/lti finding distances in the use the right triangle abc to find the length of the line segment between point a and point b. now that youve found the lengths of the legs of this right triangle, you can use the pythagorean theorem to help you find the length of the hypotenuse. thats the length of the line segment between point a and point b. start by substituting the values you know and the value you dont know into the equation. the hypotenuse is unknown, so you can use the variable c to represent the hypotenuse. the pythagorean theorem says that if a, b, and c are the sides of the right triangle, where c is the hypotenuse, then: $a^2 + b^2 = c^2$. graph of right triangle abc with points b(-1,5), c(7,5), a(7,-3) $a^2 + b^2 = c^2$ $\square^2 + 8^2 = c^2$

Explanation:

Step1: Find the length of the vertical leg

Point \( C(7,5) \) and point \( A(7, - 3) \) have the same \( x \)-coordinate, so the distance between them (length of the vertical leg) is \( |5 - (-3)|=8 \)? Wait, no, wait. Wait, the other leg: point \( B(-1,5) \) and point \( C(7,5) \) have the same \( y \)-coordinate, so the horizontal leg is \( |7 - (-1)| = 8 \). Then the vertical leg is between \( C(7,5) \) and \( A(7,-3) \), the length is \( |5-(-3)|=8 \)? Wait, no, the equation given is \( \square^{2}+8^{2}=c^{2} \). Wait, let's check the coordinates. Point \( A(7,-3) \), point \( C(7,5) \): the vertical distance is \( 5 - (-3)=8 \)? Wait, no, the \( y \)-coordinate of \( C \) is 5, \( A \) is -3, so the difference is \( 5 - (-3)=8 \). Wait, but the other leg: point \( B(-1,5) \) and \( C(7,5) \): horizontal distance is \( 7 - (-1)=8 \). Wait, but the right triangle: \( AB \) is the hypotenuse, \( BC \) is horizontal leg (length 8), \( AC \) is vertical leg. Wait, the vertical leg: from \( A(7,-3) \) to \( C(7,5) \), the length is \( 5 - (-3)=8 \)? No, wait, \( 5 - (-3)=8 \), but the square is \( 8^2 \), but the other leg: wait, no, the box is for the other leg. Wait, point \( B(-1,5) \) and \( A(7,-3) \): the horizontal difference is \( 7 - (-1)=8 \), vertical difference is \( 5 - (-3)=8 \)? No, that can't be. Wait, no, let's recalculate the vertical leg. The \( y \)-coordinate of \( C \) is 5, \( A \) is -3, so the length is \( |5 - (-3)| = 8 \). Wait, but the equation is \( \square^{2}+8^{2}=c^{2} \). Wait, maybe I made a mistake. Wait, point \( B(-1,5) \), point \( A(7,-3) \): the horizontal component is \( 7 - (-1)=8 \), vertical component is \( 5 - (-3)=8 \)? No, that would be an isoceles right triangle. Wait, but the graph: \( BC \) is from \( (-1,5) \) to \( (7,5) \), length 8. \( AC \) is from \( (7,5) \) to \( (7,-3) \), length is \( 5 - (-3)=8 \). Wait, but the equation is \( \square^{2}+8^{2}=c^{2} \). Wait, maybe the vertical leg is 8, but the other leg is also 8? No, that can't be. Wait, no, let's check the coordinates again. Point \( A(7,-3) \), point \( B(-1,5) \). The horizontal distance between \( A \) and \( B \): \( 7 - (-1)=8 \). The vertical distance: \( 5 - (-3)=8 \). Wait, so both legs are 8? But the equation is \( \square^{2}+8^{2}=c^{2} \), so the square is 8? No, that doesn't make sense. Wait, no, maybe I misread the coordinates. Wait, point \( C(7,5) \), point \( B(-1,5) \): horizontal distance is \( 7 - (-1)=8 \). Point \( C(7,5) \), point \( A(7,-3) \): vertical distance is \( 5 - (-3)=8 \). So both legs are 8. But the equation is \( \square^{2}+8^{2}=c^{2} \), so the box should be 8? No, that can't be. Wait, no, maybe the vertical leg is 8, and the horizontal leg is 8, so the box is 8. Wait, but that would be \( 8^2 + 8^2 = c^2 \). But maybe I made a mistake. Wait, let's check the coordinates again. Point \( B(-1,5) \), point \( A(7,-3) \). The \( x \)-difference: \( 7 - (-1)=8 \), \( y \)-difference: \( 5 - (-3)=8 \). So both legs are 8. Therefore, the box should be 8. Wait, but that seems odd. Wait, no, maybe the vertical leg is 8, and the horizontal leg is 8, so the first term is \( 8^2 \). Wait, the equation is \( \square^{2}+8^{2}=c^{2} \), so the square is 8. So the answer is 8.

Step1: Determine the length of the unknown leg

The vertical leg of the right triangle (between point \( A(7, - 3) \) and point \( C(7, 5) \)) has a length calculated by the difference in \( y \)-coordinates: \( 5 - (-3)=8 \). Wait, no, the horizontal leg (between point \( B(-1, 5) \) and point \( C(7, 5) \)) has length \( 7 - (-1)=8 \). The vertical leg (between \( A \) and \( C \)) has length \( 5 - (-3)=8 \). Since the Pythagorean theorem is \( a^{2}+b^{2}=c^{2} \), and one leg is 8 (given as \( 8^{2} \)), the other leg (the vertical one) is also 8. So the value in the square is 8.

Answer:

8