Answer:
√149 ≈ 12.21
Step-by-step explanation:
You want the length of the hypotenuse of the triangle with vertices at (-2, -3), (5, 7), and (5, -3).
Graph
When you graph the three points, you notice two of them are on the horizontal line y = -3, and two of them are on the vertical line x = 5. The point where those lines meet is (5, -3), the vertex where the right angle is located.
The other two points are the ends of the hypotenuse: (-2, -3) and (5, 7).
Pythagorean theorem
The horizontal leg of the triangle has a length equal to the difference of the x-coordinates of the points on the line y = -3. That length is ...
5 -(-2) = 7 . . . . units
The vertical leg of the triangle has a length equal to the difference of y-coordinates of of the points on the line x = 5. That length is ...
7 -(-3) = 10 . . . . units
The Pythagorean theorem tells you the square of the length of the hypotenuse is the sum of the squares of these horizontal and vertical lengths;
c² = a² + b²
c² = 7² +10² = 49 +100 = 149
The length of the hypotenuse is the square root of this number:
c = √149 ≈ 12.21
The approximate length of the triangle's hypotenuse is 12.21 units.
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Additional comment
If you subtract one of the hypotenuse end points from the other, you get ...
(5, 7) -(-2, -3) = (5+2, 7+3) = (7, 10)
The absolute values of these differences are the lengths of the legs of the triangle. When we use these differences together with the Pythagorean theorem, we get a formula for the distance between the two points:
[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
This formula works to compute the distance between any two points.
