Tangent of an acute angle of a rhombus is 3⁄4, whereas its side is 10 cm long. Find an area of the rhombus.
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How to find the area of the rhombus from the given data?
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To find the area of a rhombus, we need to know the lengths of its diagonals. However, in this case, we are only given the length of the side and the tangent of an acute angle.
Let's start by drawing a rhombus and labeling the given information:
A ______ B
/ \
/ \
/ \
/_____________\
D C
Side length: AB = BC = CD = DA = 10 cm
Tangent of an acute angle: tan(θ) = 3/4
Now, let's find the measure of the acute angle θ:
Using the properties of a rhombus, we know that opposite angles are equal. In this case, angle A and angle C are opposite, so θ = ∠A = ∠C.
Since we know the tangent of θ, we can use the inverse tangent (arctan) function to find the angle itself:
θ = arctan(3/4)
Using a calculator, we can find that θ ≈ 36.87 degrees.
Now that we know the measure of the angle, we can find the diagonals of the rhombus. The diagonals of a rhombus bisect each other at right angles, forming four right-angled triangles. The side length, 10 cm, is the base of these triangles, and the angle θ is one of the acute angles.
To find the length of the rhombus diagonals, we can use the trigonometric functions with the given angle and base length of the triangle.
In a right-angled triangle:
tan(θ) = opposite/adjacent
tan(θ) = length of the diagonal / (1/2 * length of the side)
length of the diagonal = tan(θ) * (1/2 * length of the side)
length of the diagonal = (3/4) * (1/2 * 10 cm) = 15/4 cm
Since the diagonals of a rhombus are perpendicular bisectors and intersect at a right angle, they divide the rhombus into four congruent right-angled triangles.
Now, we can find the height (altitude) of each triangle by using the Pythagorean theorem:
height^2 = (length of the diagonal)^2 - (1/2 * length of the side)^2
height^2 = (15/4 cm)^2 - (1/2 * 10 cm)^2
height^2 = (225/16 cm^2) - (100/4 cm^2)
height^2 = (225/16 cm^2) - (400/16 cm^2)
height^2 = (225 - 400) / 16 cm^2
height^2 = (-175) / 16 cm^2
Since height cannot be negative, we can conclude that this rhombus does not exist since the given information is not possible.
Therefore, we cannot find the area of this rhombus.