If an equilateral triangle has a perimeter of 96, what is the length of the perpendicular bisector of any of its sides?
Since the triangle is equilateral, all three sides are the same length. Let's call the length of each side "x". Therefore, the perimeter of the triangle is 3x, which is given as 96.
Therefore, x = 96 / 3 = 32.
Now, in an equilateral triangle, the perpendicular bisector of any side will form a line that passes through the opposite vertex and bisects the side at 90 degrees. This line will also divide the equilateral triangle into two congruent right triangles.
Since the perpendicular bisector bisects the side at 90 degrees, it also divides the side into two equal segments of length x/2. Using the Pythagorean Theorem, we can calculate the length of the perpendicular bisector as follows:
Let the length of the perpendicular bisector be "y".
Using the right triangle formed by the perpendicular bisector, half of the side x/2, and y, we have:
(y)^2 + (x/2)^2 = x^2
y^2 + (16)^2 = (32)^2
y^2 + 256 = 1024
y^2 = 768
y = √768
y = 27.7 (rounded to 1 decimal place)
Therefore, the length of the perpendicular bisector of any side of the equilateral triangle is approximately 27.7.
If an equilateral triangle has a perimeter of 96, what is the length of the perpendicular bisector of any of its sides?(1 point)
Responses
16√3
32
8√3
16
To find the length of the perpendicular bisector of any side of an equilateral triangle with a perimeter of 96, we can use the fact that the perimeter of an equilateral triangle can be calculated by multiplying the length of one side by 3.
Let the length of one side of the equilateral triangle be x. Therefore, the perimeter is 3x, and we are given that 3x = 96. Solving for x:
3x = 96
x = 96 / 3
x = 32
Now, in an equilateral triangle, the perpendicular bisector of any side will bisect the side into two equal parts each with length x/2 (since the triangle is equilateral). This creates a right triangle with the perpendicular bisector as the hypotenuse, and half the side length as one of the legs.
In a right triangle with one leg of length x/2, the hypotenuse is always x/2 times √3. Therefore, the length of the perpendicular bisector is:
(32/2) * √3
16 * √3
So, the length of the perpendicular bisector of any side of the equilateral triangle is 16√3.