A rectangle has a diagonal 20 inches long that forms angles of 60 and 3o with the sides. Find the perimeter of the rectangle.

To find the perimeter of the rectangle, we need to determine the length of its sides.

Let's assume the length of the rectangle is "L" inches and the width is "W" inches.

We are given that the diagonal of the rectangle is 20 inches long and forms angles of 60 degrees and 30 degrees with the sides.

Using trigonometry, we can determine the side lengths of the rectangle.

The diagonal divides the rectangle into two congruent right-angled triangles.

In the first triangle, with the 60-degree angle, the side opposite this angle (the width) is W inches.

In the second triangle, with the 30-degree angle, the side opposite this angle (the length) is L inches.

Applying the sine function to each triangle will give us:

sin(60 degrees) = W / 20 (for the first triangle)
sin(30 degrees) = L / 20 (for the second triangle)

Simplifying these equations, we get:
W = 20 * sin(60 degrees) = 20 * √3/2 = 10√3
L = 20 * sin(30 degrees) = 20 * 1/2 = 10

Now that we have the length and width, we can calculate the perimeter of the rectangle by adding the four sides:
Perimeter = 2L + 2W
Perimeter = 2(10) + 2(10√3)
Perimeter = 20 + 20√3

Therefore, the perimeter of the rectangle is 20 + 20√3 inches.

To find the perimeter of the rectangle, we first need to find the lengths of the sides. Let's denote the length of the rectangle as "l" and the width as "w".

We know that the diagonal forms angles of 60 and 30 degrees with the sides. Since the angles in a rectangle are all 90 degrees, we can use the properties of a right triangle to find the lengths of the sides.

Let's consider the right triangle formed by the width, the diagonal, and a line bisecting the angle of 30 degrees.

Using trigonometry, we can write:

sin(30 degrees) = (w/2) / 20

Simplifying, we have:

0.5 = w/40

Solving for w, we get:

w = 0.5 * 40 = 20 inches

Therefore, the width of the rectangle is 20 inches.

Now, let's consider the right triangle formed by the length, the diagonal, and a line bisecting the angle of 60 degrees.

Using trigonometry, we can write:

sin(60 degrees) = (l/2) / 20

√3/2 = l/40

Solving for l, we get:

l = (√3/2) * 40 = 20√3 inches

Therefore, the length of the rectangle is 20√3 inches.

Now that we know the length and width of the rectangle, we can calculate its perimeter.

The perimeter (P) of a rectangle is given by the formula:

P = 2 * (length + width)

Plugging in the values, we have:

P = 2 * (20√3 + 20)
P = 2 * 20√3 + 2 * 20
P = 40√3 + 40

Therefore, the perimeter of the rectangle is 40√3 + 40 inches.

the ratio of sides of a 30° - 60° - 90° triangle is

1 : √3 : 2

looks like the hypotenuse of your triangle is 10 times the basic ratio value.
so the other sides must be ........