In rhombus ABCD, the diagonals AC and BD intersect at point E. If the length of segment AE… — ACT

In rhombus $ABCD$ , the diagonals $AC$ and $BD$ intersect at point $E$ . If the length of segment $AE$ is $3$ inches and the length of segment $BE$ is $4$ inches, what is the perimeter, in inches, of the rhombus?

A
5
B
14
20Answer
D
28
E
100

Answer

The perimeter of the rhombus is 20 inches.

The diagonals of a rhombus are perpendicular and bisect each other, forming four right triangles at their intersection. In right triangle

AEB

, the legs are

AE = 3

inches and

BE = 4

inches. Using the Pythagorean theorem, the hypotenuse (which is the side

AB

of the rhombus) is

\sqrt{3^2 + 4^2} = 5

inches. Because a rhombus has four sides of equal length, the perimeter is

4 \times 5 = 20

inches.

Step-by-Step Solution

Identify the properties of the diagonals of a rhombus.

The diagonals of a rhombus are perpendicular bisectors of each other. This means they intersect at a

90^\circ

angle and divide each other into equal halves.

This allows us to model the relationship between the diagonals and the sides using right triangles.

Calculate the side length of the rhombus using the Pythagorean theorem.

In the right triangle

AEB

formed by the intersection of the diagonals, the legs are

AE = 3

inches and

BE = 4

inches. The side

AB

is the hypotenuse:

AB = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5

inches.

Knowing the side length is necessary to find the perimeter of the rhombus.

Calculate the perimeter of the rhombus.

Since all four sides of a rhombus are equal in length, the perimeter is

4 \times 5 = 20

inches.

The perimeter of any polygon is the sum of its outer boundary lengths.

Key Concept

Properties of Rhombus Diagonals and Perimeter

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