In the xy-plane, triangle ABC is a right triangle with the right angle at B. The vertices … — SAT

In the $xy$ -plane, triangle $ABC$ is a right triangle with the right angle at $B$ . The vertices of the triangle are $A(3, 11)$ , $B(15, 20)$ , and $C(27, y)$ , where $y$ is a constant. If the length of the hypotenuse $AC$ is 25 and $y < 20$ , what is the value of $y$ ?

Answer: 4

Answer

The value of

y

is 4.

Applying the distance formula to vertices

A(3, 11)

and

B(15, 20)

yields a length of 15 for side

AB

. Since triangle

ABC

has its right angle at

B

, the hypotenuse is

AC = 25

. Using the Pythagorean theorem (

AB^2 + BC^2 = AC^2

), we find that

BC = 20

. Setting the distance between

B(15, 20)

and

C(27, y)

equal to 20 gives

(27-15)^2 + (y-20)^2 = 20^2

. This simplifies to

(y-20)^2 = 256

, which has solutions

y = 36

y = 4

. Given that

y < 20

, the value of

y

must be 4.

Step-by-Step Solution

Calculate the length of segment

AB

using the distance formula.

AB = 15

To find one of the leg lengths of the right triangle.

Apply the Pythagorean theorem to find the length of leg

BC

BC = 20

To determine the length of the other leg of the right triangle.

Use the distance formula for

BC

and the condition

y < 20

to solve for

y

y = 4

To find the vertical coordinate of vertex

C

Key Concept

Using the Pythagorean theorem in coordinate geometry to solve for unknown coordinates.

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