A sign is in the shape of a rhombus with a 60° angle and sides of 2 cm long. Find the area of the sign to the nearest tenth.
area = (2)(2)sin60° = 4(√3/2) = 2√3 = appr. 3.5
To find the area of the rhombus-shaped sign, we can use the formula:
Area = (diagonal 1 * diagonal 2) / 2.
In a rhombus, the diagonals are perpendicular bisectors of each other and they divide the rhombus into four congruent right triangles.
Since the rhombus has a 60° angle, each of these right triangles has a 30-60-90 degree relationship.
The longer diagonal is twice the length of the side, so it would be 2 * 2 = 4 cm.
Using the relationship of the 30-60-90 triangle, the shorter diagonal (which is the height of the rhombus) will be 2 * sqrt(3) cm.
Now, we can plug the values into the formula:
Area = (4 cm * 2√3 cm) / 2.
Simplifying, we have:
Area = (8√3 cm) / 2.
Area = 4√3 cm.
Calculating the approximate value:
Area ≈ 4 * 1.732 cm (rounded to three decimal places).
Area ≈ 6.928 cm² (rounded to the nearest tenth).
Therefore, the area of the sign is approximately 6.9 cm².
To find the area of the rhombus, we need to use the formula A = (1/2) × d1 × d2, where A is the area and d1 and d2 are the lengths of the diagonals of the rhombus.
To determine the lengths of the diagonals, we can use trigonometry. Since we know the side length and one angle of the rhombus, we can find the lengths of the diagonals using the sine and cosine ratios.
In a rhombus, the diagonals bisect each other at right angles, splitting the rhombus into four congruent right-angled triangles. Let's consider one of these triangles.
We know that one angle of the triangle is 60° and the side opposite this angle has a length of 2 cm. We can use the sine ratio to find the length of the diagonal, as sine is the ratio of the opposite side to the hypotenuse in a right-angled triangle.
sin(60°) = opposite/hypotenuse
sin(60°) = 2/diagonal
Rearranging the equation, we have:
diagonal = 2 / sin(60°)
Using this formula, we can find the length of one of the diagonals.
diagonal = 2 / sin(60°)
diagonal ≈ 2 / 0.866 (using the value of sin(60°) from a trigonometric table or calculator)
diagonal ≈ 2.309 cm
Since the diagonals of a rhombus are congruent, both diagonals will have the same length.
Now that we know the length of one of the diagonals, we can find the other diagonal by using the fact that opposite angles in a rhombus are equal. This means that one of the interior angles opposite the 60° angle is also 60°. Therefore, we can use the same process to find the length of the other diagonal:
diagonal = 2 / sin(60°)
diagonal ≈ 2 / 0.866 (using the value of sin(60°) from a trigonometric table or calculator)
diagonal ≈ 2.309 cm
Now that we have the lengths of both diagonals (approximately 2.309 cm), we can calculate the area of the rhombus using the formula:
A = (1/2) × d1 × d2
A ≈ (1/2) × 2.309 × 2.309 (substituting the lengths of the diagonals)
A ≈ 2.665 cm²
Therefore, the area of the sign is approximately 2.7 cm² to the nearest tenth.