Question

use the right triangle abc to find the length of the line seg
you just found the length of the hypotenuse, c. that’s the lengt
and point b. label that on the coordin
(the image shows a coordinate plane with points a(8, 3), b(-2, -5), and c(8, -5), forming a right triangle. the horizontal leg from b to c is labeled 10, and the vertical leg from a to c is labeled 8.)

Explanation:

Step1: Identify the legs of the right triangle

First, we find the lengths of the two legs of the right triangle \(ABC\). For the horizontal leg (between \(B(-2, -5)\) and \(C(8, -5)\)), since the \(y\)-coordinates are the same, the length is the difference in \(x\)-coordinates: \(|8 - (-2)| = 10\). For the vertical leg (between \(A(8, 3)\) and \(C(8, -5)\)), since the \(x\)-coordinates are the same, the length is the difference in \(y\)-coordinates: \(|3 - (-5)| = 8\).

Step2: Apply the Pythagorean theorem

The Pythagorean theorem states that for a right triangle with legs \(a\) and \(b\), and hypotenuse \(c\), \(c = \sqrt{a^2 + b^2}\). Here, \(a = 10\) and \(b = 8\). So we calculate \(c = \sqrt{10^2 + 8^2} = \sqrt{100 + 64} = \sqrt{164}\). Simplifying \(\sqrt{164}\), we can factor out \(4\) (since \(164 = 4 \times 41\)): \(\sqrt{4 \times 41} = 2\sqrt{41} \approx 12.81\).

Answer:

The length of the hypotenuse (the line segment between \(A\) and \(B\)) is \(2\sqrt{41}\) (or approximately \(12.81\)).