In the xy-plane, a right triangle has vertices at the origin O(0, 0), A(x, 0), and B(0, y)… — SAT

In the $xy$ -plane, a right triangle has vertices at the origin $O(0, 0)$ , $A(x, 0)$ , and $B(0, y)$ , where $x > 0$ and $y > 0$ . If the length of the hypotenuse $AB$ is $26$ and the slope of the line containing segment $AB$ is $-\frac{12}{5}$ , what is the value of $y$ ?

Answer: 24

Answer

The value of

y

is 24.

The correct answer is 24 because the slope of the line through

A(x, 0)

and

B(0, y)

-\frac{y}{x}

. Setting this equal to

-\frac{12}{5}

gives the relationship

y = \frac{12}{5}x

. Substituting this into the Pythagorean theorem equation

x^2 + y^2 = 26^2

yields

x^2 + \left(\frac{12}{5}x\right)^2 = 676

, which simplifies to

\frac{169}{25}x^2 = 676

. Solving for

x

gives

x = 10

, and multiplying by

\frac{12}{5}

gives the value of

y

as 24.

Step-by-Step Solution

Express the slope of the line containing segment

AB

in terms of

x

and

y

y = \frac{12}{5}x

The line passes through

A(x, 0)

and

B(0, y)

, so its slope is

-\frac{y}{x}

. Setting this equal to the given slope of

-\frac{12}{5}

gives the relationship between

x

and

y

Apply the Pythagorean theorem to the right triangle

OAB

x^2 + y^2 = 676

The lengths of the legs of the right triangle are

x

and

y

, and the hypotenuse is

26

Substitute the slope relationship into the Pythagorean equation to solve for

x

x = 10

Substituting

y = \frac{12}{5}x

gives

x^2 + \frac{144}{25}x^2 = 676

, which simplifies to

\frac{169}{25}x^2 = 676

. Solving for

x

gives

10

Calculate the value of

y

y = 24

Using

x = 10

y = \frac{12}{5}x

gives

24

Key Concept

Solving right triangle problems in the coordinate plane by combining linear equations (slope) with the Pythagorean theorem.

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