A sign is in the shape of a rhombus with a 60° angle and sides of 9 cm long. Find its area to the nearest tenth
height = h
sin 60 = h/9
h = 9 sin 60
so
area = 81 sin 60 = 70.14
This didn’t help much!!
Well, it seems like you're in a "rhombus" situation here! Let's calculate the area using some hilarious math!
To find the area of a rhombus, we can use the formula: Area = (d1 * d2) / 2, where d1 and d2 are the lengths of the diagonals.
Since the rhombus has a 60° angle, we know that the diagonals will be equal. Let's find the length of the diagonals using the side length of 9 cm.
To find the length of the diagonal, let's use a bit of trigonometry! If we split the rhombus in half using one of the 60° angles, we can form a right triangle. One of the sides will be half of the given side length, which is 4.5 cm, and the hypotenuse will be the length of the diagonal.
Using the Pythagorean theorem (a^2 + b^2 = c^2), where a = 4.5 cm and b = unknown, we can solve for the length of the diagonal.
Solving for b, we get: b = sqrt(c^2 - a^2)
b = sqrt(c^2 - (4.5)^2)
b = sqrt(c^2 - 20.25)
Since both diagonals of the rhombus are equal, we can let c = d = sqrt(c^2 - 20.25)
Let's calculate the length of the diagonal by setting it equal to 9 cm and solving the equation: sqrt(c^2 - 20.25) = 9
Squaring both sides, we get: c^2 - 20.25 = 81
c^2 = 101.25
c = sqrt(101.25)
c ≈ 10.1 cm
Now that we have the length of the diagonal, we can calculate the area of the rhombus.
Area = (d1 * d2) / 2
Area = (10.1 cm * 10.1 cm) / 2
Area ≈ 51.005 cm^2
So the area of the rhombus is approximately 51.0 square centimeters. Aren't rhombuses just a "gem" of a shape?
To find the area of the rhombus, we can use the formula:
Area = (diagonal 1 * diagonal 2) / 2
First, let's find the length of the diagonals:
In a rhombus, the diagonals are perpendicular bisectors of each other, and they divide the rhombus into four congruent right-angled triangles. Since the given angle is 60°, we know that each triangle is a 30-60-90 triangle.
In a 30-60-90 triangle, the lengths of the sides are in the ratio 1:√3:2.
We are given that the sides of the rhombus are 9 cm long, so the length of the longer leg of the triangle, which is half of one of the diagonals, is 9 cm.
Using the ratio for a 30-60-90 triangle, we can find the length of the shorter leg, which is half of the other diagonal:
shorter leg = (9 cm) * (√3) / 2
Now, we can find both diagonals of the rhombus:
Diagonal 1 = 2 * shorter leg
Diagonal 2 = 2 * longer leg
Finally, we can substitute the values into the area formula:
Area = (Diagonal 1 * Diagonal 2) / 2
Calculating the values:
shorter leg = (9 cm) * (√3) / 2 ≈ 7.8 cm
longer leg = 9 cm
Diagonal 1 = 2 * shorter leg ≈ 15.6 cm
Diagonal 2 = 2 * longer leg = 18 cm
Area = (15.6 cm * 18 cm) / 2 ≈ 140.4 square cm
The area of the rhombus is approximately 140.4 square cm, rounded to the nearest tenth.