In rectangle ABCD, the length of side AD is 12 units and the length of side CD is 17 units… — ACT

In rectangle $ABCD$ , the length of side $AD$ is $12$ units and the length of side $CD$ is $17$ units. Point $E$ lies on side $CD$ such that $\triangle ADE$ is an isosceles right triangle with the right angle at vertex $D$ . What is the length, in units, of segment $BE$ ?

Answer: 13 units

Answer

The length of segment

BE

13

units.

Because

\triangle ADE

is an isosceles right triangle with the right angle at vertex

D

, leg

DE

equals leg

AD = 12

. Subtracting

DE

from total side length

CD = 17

gives segment

EC = 5

. Since

ABCD

is a rectangle, angle

C

is a right angle (

90^\circ

) and

BC = AD = 12

. Applying the Pythagorean Theorem to right triangle

\triangle BCE

gives hypotenuse

BE = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13

Step-by-Step Solution

Find the length of segment

DE

using the properties of an isosceles right triangle.

DE = 12

units

In isosceles right triangle

\triangle ADE

with right angle at

D

, legs

AD

and

DE

are equal in length. Given

AD = 12

DE

must also be

12

Determine the length of segment

EC

EC = 5

units

Since point

E

lies on side

CD

EC = CD - DE = 17 - 12 = 5

Use the Pythagorean Theorem in right triangle

\triangle BCE

to find

BE

BE = 13

units

Because

ABCD

is a rectangle, angle

C

90^\circ

and

BC = AD = 12

. Applying the Pythagorean Theorem with legs

BC = 12

and

EC = 5

gives

BE = \sqrt{12^2 + 5^2} = \sqrt{169} = 13

Key Concept

Applying properties of isosceles right triangles (

45^\circ-45^\circ-90^\circ

) and the Pythagorean Theorem in composite figures.

Question

Answer

Step-by-Step Solution

Key Concept

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