In rectangle ABCD, diagonals AC and BD intersect at point E. If the measure of \angle AEB … — ACT

In rectangle $ABCD$ , diagonals $AC$ and $BD$ intersect at point $E$ . If the measure of $\angle AEB$ is $120^\circ$ and $AC = 16$ , what is the length of side $BC$ ?

Answer: 8

Answer

The length of side

BC

8

In any rectangle, the diagonals are congruent and bisect each other. Given

AC = 16

, the distance from the intersection point

E

to any vertex is

8

, so

BE = EC = 8

. Because

\angle AEB

and

\angle BEC

lie along the straight diagonal line

AC

, they are supplementary, giving

\angle BEC = 180^\circ - 120^\circ = 60^\circ

. Triangle

BEC

is an isosceles triangle with

BE = EC = 8

and a vertex angle of

60^\circ

, which forces it to be equilateral. Consequently, all sides of

\triangle BEC

are equal, so

BC = 8

Step-by-Step Solution

Calculate the lengths of the diagonal segments from the center point

E

BE = EC = 8

The diagonals of a rectangle are congruent and bisect each other, so each half-diagonal equals half of

AC

Determine the measure of adjacent angle

\angle BEC

\angle BEC = 60^\circ

Angles

\angle AEB

and

\angle BEC

form a straight line (linear pair), so their sum is

180^\circ

Analyze

\triangle BEC

to find the length of side

BC

BC = 8

An isosceles triangle with a

60^\circ

vertex angle has base angles of

60^\circ

as well, making it an equilateral triangle where all sides are equal to

8

Key Concept

Diagonals of a rectangle are equal in length and bisect each other, dividing the rectangle into two pairs of congruent isosceles triangles.

Estimated Time:1m 15s

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