A rectangle has a diagonal of 8cm. The diagonal creates a 60 degrees angle at the base of the rectangle.
Write an exact expression for the base and the height of the rectangle.
^^ Don't get how to do that, Ive only drawn a triangular diagram, and im confused about the expression thing.
In your triangle , call your base b
and your height h,
he hypotenuse would be 8 and the base angle is 60°
so sin60° = h/8 --- h = 8sin60
cos60° = b/8 ---> b = 8cos60
yes, area = base x height
You did not say that you had to find the area
Thank you, so would I use those expressions to find the area? Would I times 8sin60 by 8 cos60?
To find the exact expressions for the base and height of the rectangle, we can use trigonometry and the properties of right triangles. Here's how you can do it:
1. Start by drawing the rectangle and its diagonal. Label the base of the rectangle as 'b' and the height as 'h'. This will help us in developing the expressions later on.
2. Connect the endpoints of the diagonal to two opposite corners of the rectangle to form a right triangle.
3. Since we have a 60-degree angle at the base of the rectangle, we can divide the right triangle into two 30-60-90 triangles.
4. In a 30-60-90 triangle, the ratio of the sides opposite the angles is as follows:
- The side opposite the 30-degree angle is half the length of the hypotenuse.
- The side opposite the 60-degree angle is √3 times the length of the side opposite the 30-degree angle.
- The hypotenuse is twice the length of the side opposite the 30-degree angle.
5. Since the diagonal of the rectangle is 8 cm, it is also the hypotenuse of the right triangle. Therefore, the side opposite the 30-degree angle is half the length of the diagonal, which is 4 cm.
6. Now, applying the ratios mentioned above, we can determine the expression for the height of the rectangle:
- The height, 'h,' is the side opposite the 60-degree angle. Since it is √3 times the side opposite the 30-degree angle, we have h = √3 * 4 = 4√3 cm.
7. Similarly, we can determine the expression for the base of the rectangle:
- The base, 'b,' is the side adjacent to the 60-degree angle in the right triangle. It is equal to the side opposite the 30-degree angle, which is 4 cm.
Therefore, the exact expressions for the base and height of the rectangle are:
Base (b) = 4 cm
Height (h) = 4√3 cm
Remember, these expressions hold true for the given conditions of a rectangle with a diagonal of 8 cm and a 60-degree angle at the base.