Examkin

Question

Difficulty: MediumProperties of Quadrilaterals

A rhombus has a perimeter of 40 centimeters and one diagonal of length 12 centimeters. What is the area, in square centimeters, of the rhombus?

  1. A
    48
  2. B
    60
  3. C
    120
  4. D
    192
  5. 96Answer

Answer

96 square centimeters
Because all four sides of a rhombus are congruent, a perimeter of 40 centimeters gives a side length of 10 centimeters. The diagonals of a rhombus intersect at right angles and bisect each other. Half of the given diagonal is 6 centimeters. By applying the Pythagorean theorem (62+b2=1026^2 + b^2 = 10^2), the second half-diagonal is found to be 8 centimeters, which means the full length of the second diagonal is 16 centimeters. Using the rhombus area formula Area=12⋅d1⋅d2\text{Area} = \frac{1}{2} \cdot d_1 \cdot d_2, the area is 12⋅12⋅16=96\frac{1}{2} \cdot 12 \cdot 16 = 96 square centimeters.

Step-by-Step Solution

1
Calculate the side length of the rhombus from its perimeter.
Side length s=404=10 cms = \frac{40}{4} = 10\text{ cm}.
All four sides of a rhombus are equal in length.
2
Use the properties of rhombus diagonals to find the length of the second diagonal.
Half of the given diagonal is 122=6 cm\frac{12}{2} = 6\text{ cm}. In the right triangle formed by the half-diagonals and a side: 62+b2=102  ⟹  36+b2=100  ⟹  b=8 cm6^2 + b^2 = 10^2 \implies 36 + b^2 = 100 \implies b = 8\text{ cm}. Thus, the second diagonal d2=2×8=16 cmd_2 = 2 \times 8 = 16\text{ cm}.
The diagonals of a rhombus are perpendicular bisectors of each other.
3
Calculate the area of the rhombus using the diagonal formula.
\text{Area} = \frac{1}{2} \times d_1 \times d_2 = \frac{1}{2} \times 12 \times 16 = 96\text{ sq cm}.
The area of any rhombus is equal to half the product of the lengths of its diagonals.

Key Concept

Properties of Rhombus Diagonals and Area Calculation
Rate this question

Topics

Properties of Quadrilaterals

More questions from this topic