the larger angles of a rhombus are twice the smaller angle of the rhombus. if the shorter diagonal is 20, find the perimeter of the rhombus
since consecutive angles total 180, the angles are 60 and 120
Now draw a diagram and I'm sure you will see all the 30-60-90 right triangles in there.
To find the perimeter of a rhombus, we first need to find the length of one side.
Let's assume that the smaller angle of the rhombus is "x" degrees. According to the given condition, the larger angles are twice the measure of the smaller angle, so the larger angles measure 2x degrees.
In a rhombus, the diagonals are perpendicular bisectors of each other, meaning they split each other into two equal parts. Therefore, each diagonal is divided into two segments of equal length.
We are given that the shorter diagonal is 20 units long. Since the diagonals of a rhombus are perpendicular bisectors of each other, we can use the Pythagorean theorem to find the length of each segment of the shorter diagonal.
Let's assume that each segment of the shorter diagonal measures "d" units. Using the Pythagorean theorem, we have:
d² + d² = 20²
2d² = 400
d² = 200
d ≈ √200
d ≈ 14.14
Now, we know that each segment of the shorter diagonal measures approximately 14.14 units long. Since the diagonals of a rhombus form four congruent right-angled triangles, we can find the lengths of the sides using trigonometry.
In each right-angled triangle, the hypotenuse is one of the sides of the rhombus (which we are looking for), and one of the other sides is half the length of the shorter diagonal (d/2).
Using the Pythagorean theorem, we can find the length of the remaining side:
s² = (d/2)² + (d/2)²
s² = (14.14/2)² + (14.14/2)²
s² = 7.07² + 7.07²
s² = 100
s = 10
Therefore, each side of the rhombus measures approximately 10 units.
Now we can find the perimeter of the rhombus by multiplying the length of one side (10 units) by 4 (since a rhombus has four equal sides):
Perimeter = 4 * 10
Perimeter = 40 units
Thus, the perimeter of the given rhombus is 40 units.