In quadrilateral ABCD, \angle B and \angle D are right angles. The lengths of three sides … — Armed Services Vocational Aptitude Battery (ASVAB)

In quadrilateral $ABCD$ , $\angle B$ and $\angle D$ are right angles. The lengths of three sides are given as $AB = 7$ , $BC = 24$ , and $CD = 15$ . What is the length of side $AD$ ?

A
8
B
10
20Answer
D
46

Answer

The length of side

AD

is 20.

Drawing the diagonal

AC

creates two right triangles that share a hypotenuse. The first triangle (

ABC

) has legs of

7

and

24

, giving a hypotenuse of

25

(

7^2 + 24^2 = 25^2

). In the second triangle (

ADC

), this

25

serves as the hypotenuse while

CD

is a leg of

15

. Solving for the remaining leg gives

20

(

25^2 - 15^2 = 400

, and

\sqrt{400} = 20

Step-by-Step Solution

Visualize or draw diagonal

AC

to split the quadrilateral into two right triangles.

Right

\triangle ABC

and right

\triangle ADC

are formed, sharing hypotenuse

AC

Since

\angle B

and

\angle D

are right angles, the segment connecting

A

and

C

must be the hypotenuse for both triangles.

Apply the Pythagorean theorem to

\triangle ABC

to find the length of

AC

AC^2 = 7^2 + 24^2 = 49 + 576 = 625

, which means

AC = 25

The lengths of both legs (

AB

and

BC

) are known, allowing calculation of the hypotenuse.

Apply the Pythagorean theorem to

\triangle ADC

using the known hypotenuse.

AD^2 + CD^2 = AC^2

, which becomes

AD^2 + 15^2 = 25^2

The hypotenuse

AC

is shared, and leg

CD

is given as 15, leaving

AD

as the only unknown variable.

Solve the equation for the missing leg

AD

AD^2 + 225 = 625 \Rightarrow AD^2 = 400 \Rightarrow AD = 20

Isolating

AD^2

and taking the square root provides the final length of the side.

Key Concept

Applying the Pythagorean theorem sequentially across adjacent right triangles that share a common hypotenuse.

Estimated Time:2m 0s

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