In the figure, triangle RST is a right triangle with the right angle at vertex S. If the l… — SAT

In the figure, triangle $RST$ is a right triangle with the right angle at vertex $S$ . If the length of side $RS$ is $8$ and the length of side $ST$ is $15$ , what is the length of the hypotenuse $RT$ ?

A
7
B
13
17Answer
D
23

Answer

According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the legs:

RT^2 = RS^2 + ST^2

. Substituting the given values yields

RT^2 = 8^2 + 15^2 = 64 + 225 = 289

. Taking the square root of

289

gives

RT = 17

Step-by-Step Solution

Set up the Pythagorean theorem for right triangle

RST

RS^2 + ST^2 = RT^2

In any right triangle, the sum of the squares of the legs equals the square of the hypotenuse.

Substitute the given side lengths into the equation.

8^2 + 15^2 = RT^2

The legs of the triangle are

RS = 8

and

ST = 15

Calculate the squares and sum them.

64 + 225 = 289 = RT^2

8^2 = 64

and

15^2 = 225

. Adding them gives

289

Take the square root of both sides to find the length of the hypotenuse.

RT = \sqrt{289} = 17

The length must be a positive value, and

\sqrt{289}

simplifies to

17

Key Concept

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs:

a^2 + b^2 = c^2

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