If the height of an equilateral triangle is 9, what is its area?
F. 27
G. 54
H. 81
J. 27 sqrt(3)
K. 54 sqrt (3)
A = h ^ 2 / sqrt ( 3 )
A = 9 ^ 2 / sqrt ( 3 )
A = 81 / sqrt ( 3 )
A = 27 * 3 / sqrt ( 3 )
A = 27 * sqrt ( 3 ) * sqrt ( 3 ) / sqrt ( 3 )
A = 27 * sqrt ( 3 )
Answer J
Thank you so much, sir.
Thank you so much, sir. for answering
To find the area of an equilateral triangle, we need to know the length of one of its sides or the height of the triangle. In this case, the question states that the height of the equilateral triangle is 9.
The height of an equilateral triangle divides it into two congruent right triangles, as shown below:
/|\
/ | \
h / | \
/ | \
/____|____\
In this diagram, h represents the height, and the sides of the equilateral triangle are equal in length.
Since the triangle is equilateral, each of its angles measures 60 degrees. Thus, the right angle in each right triangle is half of that, or 30 degrees.
Now, the base of each right triangle is equal to half of the side length of the equilateral triangle. Let's call this value s.
Using trigonometry, we can say that:
sin(30 degrees) = h / s
Since the sine of 30 degrees is equal to 1/2, we can rearrange the equation to solve for s:
1/2 = h / s
s = 2h
So, the length of each side of the equilateral triangle is equal to twice its height, which in this case is 2 * 9 = 18.
Now, we can calculate the area of the equilateral triangle using the formula:
Area = (sqrt(3) / 4) * s^2
Substituting the value of s, we get:
Area = (sqrt(3) / 4) * (18^2)
= (sqrt(3) / 4) * 324
= sqrt(3) * 81
Therefore, the area of the equilateral triangle is 81 * sqrt(3).
Hence, the correct answer is option H. 81.