A right triangle is plotted in the standard (x, y) coordinate plane. The vertices of the t… — ACT

A right triangle is plotted in the standard $(x, y)$ coordinate plane. The vertices of the triangle are located at $(-3, -2)$ , $(5, -2)$ , and $(-3, 4)$ . What is the length of the hypotenuse of the triangle?

A
$\frac{4}{3}$
B
$2\sqrt{7}$
C
$\frac{3}{5}$
$10$ Answer
E
$14$

Answer

The distance between the vertices

(-3, -2)

and

(5, -2)

along the horizontal line

y = -2

5 - (-3) = 8

units. The distance between the vertices

(-3, -2)

and

(-3, 4)

along the vertical line

x = -3

4 - (-2) = 6

units. Because horizontal and vertical segments meet at a right angle, they form the legs of a right triangle. Applying the Pythagorean theorem, the length of the hypotenuse is the square root of the sum of the squares of the legs:

\sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10

Step-by-Step Solution

Determine the lengths of the two perpendicular legs of the right triangle on the coordinate plane.

The horizontal leg has a length of

5 - (-3) = 8

units, and the vertical leg has a length of

4 - (-2) = 6

units.

Because the segment between

(-3, -2)

and

(5, -2)

is horizontal (constant

y = -2

) and the segment between

(-3, -2)

and

(-3, 4)

is vertical (constant

x = -3

Apply the Pythagorean theorem to calculate the length of the hypotenuse.

The hypotenuse length is

\sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10

units.

The Pythagorean theorem states that for any right triangle with legs

a

and

b

and hypotenuse

c

a^2 + b^2 = c^2

Key Concept

Finding the lengths of segments and applying the Pythagorean theorem on the coordinate plane.

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