A rectangle has a diagonal of 18cm. The diagonal creates a 60 degree angle at the base of the triangle. Write an exact expression for the base and the height of the triangle.
I'm a little rusty, but here goes...
In your triangle , call your base b & your height h, the hypotenuse would be 18 & the base angle is 60°
so sin60° = h/18 ---> h = 18sin60
cos60° = b/18 ---> b = 18cos60
So, later when you need area a = b*h
To find the base and height of the rectangle, we can use the Pythagorean Theorem. In a rectangle, the diagonal is the hypotenuse of a right triangle formed by the base and height.
Let's assume that the base of the rectangle is "b" and the height is "h". Then, we can set up the Pythagorean equation as follows:
b^2 + h^2 = 18^2
We also know that the diagonal creates a 60 degree angle at the base of the triangle. By drawing a line connecting the midpoint of the diagonal with the opposite corner of the rectangle, we can see that this line bisects the rectangle into two congruent right triangles.
Since we have a 60 degree angle, we know that the triangle is a special right triangle, specifically a 30-60-90 triangle. In a 30-60-90 triangle, the ratio of the sides opposite the angles is 1:√3:2.
In our case, the base of the triangle is equal to the height of the rectangle, so b = h.
Using the ratios of the 30-60-90 triangle, we can express the base and height of the rectangle as:
base = b = h
height = h = √3b
Substituting these expressions into the Pythagorean equation:
b^2 + (√3b)^2 = 18^2
b^2 + 3b^2 = 324
4b^2 = 324
b^2 = 81
b = √81
b = 9 cm
Since the base and height are equal, the base and height of the rectangle are both equal to 9 cm.