A diagonal of a rectangle forms an angle of 60 degree with each of the two shorter sides of the rectangle.If the length of a shorter side of the rectangle is 3,what is the length of the diagonal?
Let the lenght of the diagonal be x
sin60=3/x
To find the length of the diagonal of a rectangle, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (in this case, the diagonal) is equal to the sum of the squares of the other two sides.
In this problem, the diagonal of the rectangle forms a right triangle with the two shorter sides. We are given that the angle formed between the diagonal and each of the shorter sides is 60 degrees. Since the two shorter sides are equal in length (since they are sides of a rectangle), let's call the length of each shorter side "x".
Using trigonometric ratios, we know that the sine of an angle in a right triangle is equal to the length of the side opposite that angle divided by the length of the hypotenuse. In this case, the length of each shorter side is opposite to the angle of 60 degrees, and the diagonal is the hypotenuse.
Sine 60 degrees = Opposite side / Hypotenuse
sin(60) = x / diagonal
Now, let's solve for the length of the diagonal. We'll rearrange the equation and substitute the value of x as 3 (since we are given that the length of each shorter side is 3):
diagonal = x / sin(60)
diagonal = 3 / sin(60)
To evaluate sin(60), we can use a calculator or reference tables. The value of sin(60) is √3/2.
diagonal = 3 / (√3/2)
diagonal = (3 * 2) / √3
diagonal = 6 / √3
To rationalize the denominator, we multiply both the numerator and denominator by √3 (the conjugate of √3/√3):
diagonal = (6 / √3) * (√3 / √3)
diagonal = 6√3 / 3
diagonal = 2√3
Therefore, the length of the diagonal of the rectangle is 2√3.