The length of a rectangle is 142.9 in. If the diagonal makes an angle of 32.1o with this side, find the measure of the diagonal of the rectangle rounded to the nearest tenth of an inch. let x represent the diagonal.
cos 32.1 = side/diagonal = 142.9/x
so
x = 142.9 / cos 32.1
To find the measure of the diagonal of the rectangle, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (diagonal) is equal to the sum of the squares of the lengths of the other two sides.
In this case, the diagonal forms a right triangle with the length of the rectangle (142.9 in) and one of the sides. The angle given (32.1 degrees) helps us identify the side that forms the right angle.
Let's call the side adjacent to the angle as the base, and the side opposite to the angle as the height. Since the hypotenuse is the diagonal, we have:
cos(32.1) = base / diagonal
Now we can rearrange the equation to solve for the diagonal:
diagonal = base / cos(32.1)
Now substituting the values we have:
base = 142.9 in
angle = 32.1 degrees
diagonal = 142.9 in / cos(32.1)
Using a calculator, evaluate cos(32.1) which is approximately 0.8436.
diagonal = 142.9 in / 0.8436
diagonal ≈ 169.4 in
Therefore, the measure of the diagonal of the rectangle, rounded to the nearest tenth of an inch, is approximately 169.4 inches.