In quadrilateral ABCD, \angle B = 90^\circ and \angle D = 90^\circ. If AB = 12 units, BC =… — ACT

In quadrilateral $ABCD$ , $\angle B = 90^\circ$ and $\angle D = 90^\circ$ . If $AB = 12$ units, $BC = 16$ units, and $AD = 10$ units, what is the length, in units, of side $CD$ ?

$10\sqrt{3}$ Answer
B
$10\sqrt{5}$
C
$6\sqrt{19}$
D
$18$
E
$10$

Answer

10\sqrt{3}

units

Connecting vertex

A

to vertex

C

creates two right-angled triangles sharing hypotenuse

AC

. In right triangle

ABC

, the Pythagorean theorem yields

AC^2 = 12^2 + 16^2 = 400

, so

AC = 20

. Next, in right triangle

ADC

AC

serves as the hypotenuse and

AD = 10

is one leg. Solving for leg

CD

gives

CD^2 = AC^2 - AD^2 = 20^2 - 10^2 = 300

, which simplifies to

CD = \sqrt{300} = 10\sqrt{3}

Step-by-Step Solution

Draw diagonal

AC

to divide quadrilateral

ABCD

into two right triangles,

\triangle ABC

and

\triangle ADC

, sharing hypotenuse

AC

Two right-angled triangles

\triangle ABC

(with right angle at

B

) and

\triangle ADC

(with right angle at

D

Diagonal

AC

acts as the hypotenuse for both right triangles.

Apply the Pythagorean Theorem to right triangle

\triangle ABC

to calculate the length of hypotenuse

AC

AC = \sqrt{AB^2 + BC^2} = \sqrt{12^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20

units.

In right triangle

\triangle ABC

AB

and

BC

are legs.

Apply the Pythagorean Theorem to right triangle

\triangle ADC

to solve for leg

CD

CD = \sqrt{AC^2 - AD^2} = \sqrt{20^2 - 10^2} = \sqrt{400 - 100} = \sqrt{300} = 10\sqrt{3}

units.

In right triangle

\triangle ADC

AC

is the hypotenuse (

20

) and

AD

is a leg (

10

Key Concept

Multi-step applications of the Pythagorean Theorem using shared boundary hypotenuses.

Estimated Time:1m 15s

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