A rhombus has diagonals of 32 and 20 inches. Determine its area.
The correct answer and explanation is :
To find the area of a rhombus, we use the formula:
[
\text{Area} = \frac{1}{2} \times d_1 \times d_2
]
where:
- ( d_1 ) is the length of the first diagonal,
- ( d_2 ) is the length of the second diagonal.
In the problem, we are given:
- ( d_1 = 32 ) inches (the first diagonal),
- ( d_2 = 20 ) inches (the second diagonal).
Step 1: Apply the formula
Substitute the values of the diagonals into the area formula:
[
\text{Area} = \frac{1}{2} \times 32 \times 20
]
Step 2: Calculate the result
First, multiply the diagonals:
[
32 \times 20 = 640
]
Then, divide the product by 2:
[
\frac{640}{2} = 320
]
Thus, the area of the rhombus is ( 320 ) square inches.
Explanation
A rhombus is a type of parallelogram where all sides have equal length, but the angles between them can vary. One key property of a rhombus is that its diagonals bisect each other at right angles (90 degrees). This means that the diagonals divide the rhombus into four right-angled triangles.
The formula for the area of a rhombus involves its diagonals because the area can be thought of as the sum of the areas of these triangles. The diagonals create four congruent right-angled triangles, and the area of each triangle can be found using the base and height (which are half the lengths of the diagonals). Multiplying the lengths of the diagonals gives the total area covered by all four triangles, and dividing by 2 accounts for the fact that the diagonals cut the rhombus into two equal parts.
Thus, using the given diagonal lengths, we can directly compute the area as 320 square inches.