**EDIT**
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.
My teacher wants the base and height answered in roots so do you think you could switch the answers up? You need to use sin, cos, or tan 60 to find the answer and I don't know how. Ms. H, can you help me?
Are you currently using Pythagorean theorem in class?
OK...lets name the diagonal-d, the base - b and the height-h, and use the Pythagorean theorem: a^2 + b^2 = c^2 c=hypotenuse
(We will substitute the base our letters in.)
b^2 + h^2 = d^2
h = √(d^2 - b^2)
..and...
b = √(d^2 - h^2)
Let me know if this is the format. I used the Pythagorean's Theorem, but you need another side length in order to solve the third side.
Of course, I can help you with that! To find the exact expression for the base and height of the triangle, we need to use trigonometric ratios such as sine, cosine, or tangent.
First, let's label the triangle. Let's call the base of the triangle 'b', and the height 'h'. The diagonal of the rectangle creates a right-angled triangle, so we can use trigonometry to relate the side lengths.
In this case, we are given that the diagonal has a length of 18 cm and creates a 60-degree angle at the base of the triangle. The 60-degree angle is opposite to the height of the triangle, which means we can use the sine ratio.
The sine ratio states that the ratio of the length of the side opposite an angle to the hypotenuse is equal to the sine of that angle. In this case, the height 'h' is opposite to the 60-degree angle, and the hypotenuse is the diagonal of the rectangle, which is 18 cm.
Using the sine ratio, we have:
sin(60) = h / 18
To find the exact expression for 'h', we can rearrange the equation:
h = 18 * sin(60)
Now, let's calculate the value. The sine of 60 degrees is (√3) / 2, so we have:
h = 18 * (√3) / 2
h = 9√3 cm
Therefore, the height of the triangle is 9√3 cm.
Next, let's find the base 'b'. Since we know the diagonal of the rectangle, we can use the Pythagorean theorem to relate the base, height, and diagonal.
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
In this case, the hypotenuse is the diagonal (18 cm), the base is 'b', and the height is 'h' (9√3 cm).
Using the Pythagorean theorem, we have:
b^2 + (9√3)^2 = 18^2
Simplifying the equation:
b^2 + 243 = 324
b^2 = 324 - 243
b^2 = 81
Taking the square root of both sides:
b = √81
b = 9 cm
Therefore, the base of the triangle is 9 cm.
To summarize:
- The height of the triangle is 9√3 cm.
- The base of the triangle is 9 cm.
Please let me know if you need any further explanation or assistance!