A sign is in the shape of a rhombus with a 60° angle and sides of 7 cm long. Find its area to the nearest tenth.
the height is 7 sin60°
and as usual, area = length * height
To find the area of a rhombus, you can use the formula: Area = (diagonal1 × diagonal2) / 2.
In this case, the given information is the measure of one angle (60°) and the length of one side (7 cm).
To find the diagonals of the rhombus, we can use the properties of a rhombus. In a rhombus, opposite angles are equal, so we know that there are two 60° angles.
Since the sum of all angles in a rhombus is 360°, we can calculate the measure of the remaining two angles by subtracting 2 × 60° from 360°. That gives us 240°, so each of the remaining angles is 120°.
A rhombus has diagonals that bisect each other at right angles. Therefore, the opposite angles of the rhombus will form right triangles with the diagonals. The given angle (60°) is one of the acute angles of this right triangle.
Now we can use trigonometric functions to find the length of the diagonals. We will use the sine function to find the lengths of the diagonals.
For the given angle (60°) and the known side length of 7 cm, we can use the sine function to calculate the length of the diagonal using the formula:
sin(angle) = opposite/hypotenuse.
In this case, the opposite side is half of the diagonal, and the hypotenuse is the length of the side (7 cm). So we have:
sin(60°) = (1/2) × diagonal / 7 cm.
Re-arranging the formula, we find:
diagonal = (sin(60°) × 7 cm) / (1/2).
Using a calculator, we determine:
diagonal ≈ 8.08 cm.
Since the diagonals of a rhombus are equal in length, both diagonals of the rhombus are approximately 8.08 cm.
Now, substituting the values into the formula for the area of a rhombus, we have:
Area = (8.08 cm × 8.08 cm) / 2.
Calculating this, we find:
Area ≈ 32.74 cm².
Therefore, the area of the rhombus, rounded to the nearest tenth, is approximately 32.7 cm².