A right triangle in the xy-plane has vertices at the coordinates (0, 0), (15, 0), and (15,… — SAT

A right triangle in the $xy$ -plane has vertices at the coordinates $(0, 0)$ , $(15, 0)$ , and $(15, 8)$ . What is the perimeter of this triangle?

A
17
B
23
40Answer
D
60

Answer

The perimeter of the triangle is 40.

The perimeter of the triangle is the sum of its three side lengths. The horizontal leg has a length of

15

(from

(0,0)

(15,0)

), and the vertical leg has a length of

8

(from

(15,0)

(15,8)

). Using the Pythagorean theorem, the hypotenuse has a length of

\sqrt{15^2 + 8^2} = \sqrt{225 + 64} = 17

. Summing these three lengths gives a perimeter of

15 + 8 + 17 = 40

Step-by-Step Solution

Identify the lengths of the two perpendicular legs of the right triangle from the given coordinates.

The horizontal leg lies on the x-axis from

(0, 0)

(15, 0)

, which has a length of

15 - 0 = 15

. The vertical leg runs parallel to the y-axis from

(15, 0)

(15, 8)

, which has a length of

8 - 0 = 8

The length of horizontal and vertical segments in a coordinate plane can be found by taking the absolute difference of their non-matching coordinates.

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

The hypotenuse length is

\sqrt{15^2 + 8^2} = \sqrt{225 + 64} = \sqrt{289} = 17

The Pythagorean theorem states that

a^2 + b^2 = c^2

for any right triangle with legs

a

and

b

and hypotenuse

c

Calculate the perimeter of the triangle by summing the lengths of all three sides.

15 + 8 + 17 = 40

The perimeter of a triangle is the total distance around the boundary, which equals the sum of its three side lengths.

Key Concept

Calculating the perimeter of a right triangle in the coordinate plane by using the Pythagorean theorem to find the hypotenuse.

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