The diagonals of a rectangle are 12 inches long and intersect at an angle of 60 degrees. Find the perimeter of the rectangle.
The angle bet. the diag. and the long
side = 30 deg.
X = 12cos30 = 10.4in = l0ng side.
Y = 12sin30 = 6in = short side.
P = 2*10.4 + 2*6 = 32.8in.
To find the perimeter of the rectangle, we need to find the length and width of the rectangle.
Let's label the length of the rectangle as "L" and the width as "W".
Since the diagonals of a rectangle bisect each other, the length of one of the diagonals is equal to half of the rectangle's diagonal.
Given that the length of the diagonal is 12 inches, the length of one of the diagonals is 12/2 = 6 inches.
Since the length and width of a rectangle form a right triangle with the diagonal as the hypotenuse, we can use the Pythagorean theorem to find L and W.
From the right triangle formed by L, W, and the diagonal, we have:
L^2 + W^2 = diagonal^2
Substituting the values we have, we get:
L^2 + W^2 = 6^2
L^2 + W^2 = 36
Since the diagonals of a rectangle are equal in length, we know that:
L = W
Substituting L for W in the equation above, we get:
L^2 + L^2 = 36
2L^2 = 36
Dividing both sides of the equation by 2, we get:
L^2 = 18
Taking the square root of both sides of the equation, we get:
L = √18
Simplifying, we get:
L ≈ 4.24 inches (rounded to two decimal places)
Now that we have the length of the rectangle, we can find the width by substituting the value of L into the equation L = W:
4.24 = W
Therefore, the width of the rectangle is also approximately 4.24 inches.
To find the perimeter, we add up all four sides:
Perimeter = 2(L + W + L + W)
Substituting the values we have, we get:
Perimeter = 2(4.24 + 4.24 + 4.24 + 4.24)
Perimeter = 2(16.96)
Perimeter = 33.92 inches
Therefore, the perimeter of the rectangle is approximately 33.92 inches.
To find the perimeter of the rectangle, we first need to find the lengths of its sides.
Let's label the sides of the rectangle as 'a' and 'b'.
Since the diagonals of a rectangle bisect each other and form right angles, we can consider half of one diagonal as the hypotenuse of a right triangle.
Let's consider half of one of the diagonals as the hypotenuse. Therefore, the length of this hypotenuse is 6 inches (half of 12 inches).
Now, let's consider the vertical and horizontal sides of the rectangle formed by the diagonals. These sides are the legs of the right triangle.
Since the diagonals of the rectangle intersect at an angle of 60 degrees, the angle formed by one of the legs with the hypotenuse is 30 degrees (since the diagonals bisect each other).
Let's label one leg of the right triangle as 'a' and the other leg as 'b'. We are trying to find the lengths of these legs.
Applying trigonometric ratios, we can say:
sin(30 degrees) = a / (6 inches)
Rearranging the equation, we get:
a = 6 inches * sin(30 degrees)
Using a calculator, sin(30 degrees) is equal to 0.5.
So, a = 6 inches * 0.5 = 3 inches.
Since the rectangle is symmetrical, both a and b have the same length of 3 inches.
Therefore, the lengths of the sides of the rectangle are 3 inches and 3 inches.
To find the perimeter, we sum up all the sides:
Perimeter = 2a + 2b = 2(3 inches) + 2(3 inches) = 12 inches.
Thus, the perimeter of the rectangle is 12 inches.