Question

use the right triangle abc to find the length of the line segment between point a and point b. you can think of the distance between point a and point b as the length of a hypotenuse of a right triangle. start by finding the length of the legs of this right triangle. what is the length of the line segment between point a and point c? label the length on the coordinate plane. the coordinate plane has point a (8, 3), point b (-2, -5), point c (8, -5).

Explanation:

Step1: Find length of AC

Points A(8, 3) and C(8, -5) have same x - coordinate. The distance formula for vertical line is $|y_2 - y_1|$. So length of AC is $|3 - (-5)| = |3 + 5| = 8$.

Step2: Find length of BC

Points B(-2, -5) and C(8, -5) have same y - coordinate. The distance formula for horizontal line is $|x_2 - x_1|$. So length of BC is $|8 - (-2)| = |8 + 2| = 10$.

Step3: Find length of AB (using Pythagorean theorem)

In right triangle ABC, $AB^2 = AC^2 + BC^2$. Substitute AC = 8 and BC = 10. So $AB^2 = 8^2 + 10^2 = 64 + 100 = 164$. Then $AB=\sqrt{164}=2\sqrt{41}\approx12.81$ (or using distance formula: $AB=\sqrt{(8 - (-2))^2+(3 - (-5))^2}=\sqrt{(10)^2+(8)^2}=\sqrt{100 + 64}=\sqrt{164}=2\sqrt{41}$). Also, length of AC is 8.

Answer:

Length of AC: 8; Length of AB: $2\sqrt{41}$ (or approximately 12.81)