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Difficulty: MediumQuadrilaterals and Polygons

In rhombus ABCDABCD, the perimeter is 5252 and the length of diagonal ACAC is 1010. What is the area of rhombus ABCDABCD?

  1. A
    60
  2. B
    96
  3. 120Answer
  4. D
    169
  5. E
    240

Answer

The area of rhombus ABCDABCD is 120.
Because all four sides of a rhombus are equal in length, a perimeter of 52 implies each side measures 13. The diagonals of a rhombus intersect at right angles and bisect each other. Given diagonal AC has a length of 10, half of AC is 5. Using the Pythagorean theorem on one of the right triangles formed by the intersecting diagonals gives a half-diagonal length of sqrt(13^2 - 5^2) = 12 for BD. Thus, the total length of diagonal BD is 24. The area of the rhombus is (1/2) * 10 * 24 = 120.

Step-by-Step Solution

1
Find the side length of rhombus ABCDABCD.
Side length s=52/4=13s = 52 / 4 = 13.
A rhombus has four equal sides, so its perimeter divided by 4 gives the length of one side.
2
Use the properties of rhombus diagonals to find half of diagonal BDBD.
Half of AC=5AC = 5. Half of BD=132−52=169−25=144=12BD = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12.
The diagonals of a rhombus bisect each other at right angles, forming four congruent right triangles whose hypotenuse is the side length (1313) and legs are the half-diagonals.
3
Calculate the full length of diagonal BDBD.
Length of diagonal BD=2×12=24BD = 2 \times 12 = 24.
The full diagonal length is twice the length of its half.
4
Calculate the area of rhombus ABCDABCD.
Area =12×d1×d2=12×10×24=120= \frac{1}{2} \times d_1 \times d_2 = \frac{1}{2} \times 10 \times 24 = 120.
The area of any rhombus is equal to half the product of its diagonal lengths.

Key Concept

Properties of a Rhombus: Perpendicular Bisecting Diagonals and Area Formula
Estimated Time:1m 30s
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Topics

Quadrilaterals and Polygons

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