If the diagonal of a rhombus is equal to a side, find the angles of the rhombus.
Let's label the length of a side of the rhombus as "s".
Since the diagonal of the rhombus is equal to a side, we can form two congruent right triangles by drawing the diagonal. Let's denote the diagonal as "d".
In each right triangle, the length of the hypotenuse (d) is equal to the length of a side of the rhombus (s), and one of the legs is the length of half a side of the rhombus (s/2).
We can apply the Pythagorean theorem to one of the right triangles:
(d/2)^2 + (s/2)^2 = s^2
(d^2)/4 + (s^2)/4 = s^2
(d^2 + s^2)/4 = s^2
d^2 + s^2 = 4s^2
d^2 = 3s^2
From this equation, we know that the diagonal squared is equal to 3 times the side length squared.
Now, let's find the angles of the rhombus.
In a rhombus, opposite angles are congruent. Let's denote one of the angles as "θ".
We can apply the Law of Cosines to one of the congruent triangles:
s^2 = (s/2)^2 + (s/2)^2 - 2(s/2)(s/2)cos(θ)
s^2 = s^2/2 - 2(s^2)/4 cos(θ)
s^2 = s^2 - (s^2)/2 cos(θ)
s^2/2 = (s^2)/2 cos(θ)
cos(θ) = 1/2
In a rhombus, all four angles are congruent. Therefore, all four angles of the rhombus measure:
θ = arccos(1/2)
Using a calculator, we find:
θ ≈ 60 degrees
So, the angles of the rhombus measure approximately 60 degrees each.