how do i get the answer to this question. the base of an isosceles triangle is half as the two equal sides. write the area of the triangle as a function of the length of the base
To find the area of an isosceles triangle as a function of the length of the base, we need to use the formula for the area of a triangle.
The formula for the area of a triangle is:
Area = 1/2 * base * height
In this case, the given information states that the base of the isosceles triangle is half the length of the two equal sides. Let's assume the length of the equal sides is "s". Therefore, the length of the base will be "s/2".
Now, we need to find the height of the triangle. The height is a line perpendicular to the base and connecting the base to the top vertex (also known as the apex). Since this is an isosceles triangle, the height will divide the triangle into two congruent right-angled triangles.
To find the height, we can use the Pythagorean theorem. The Pythagorean theorem states that for a right-angled triangle, the sum of the squares of the two shorter sides (legs) is equal to the square of the longest side (hypotenuse). In our case, the legs are the height and half of the base, and the hypotenuse is the equal side.
So, using the Pythagorean theorem, we can write:
(height)^2 + (s/2)^2 = s^2
Simplifying this equation, we get:
height^2 = s^2 - (s/2)^2
height^2 = 4s^2/4 - s^2/4
height^2 = (3s^2)/4
Taking the square root of both sides, we have:
height = sqrt((3s^2)/4)
height = (s * sqrt(3)) / 2
Now, substitute the values for the base and height into the area formula:
Area = 1/2 * base * height
= 1/2 * (s/2) * ((s * sqrt(3)) / 2)
= (s^2 * sqrt(3)) / 8
Therefore, the area of an isosceles triangle with a base that is half the length of the two equal sides can be expressed as the function:
Area(s) = (s^2 * sqrt(3)) / 8, where 's' represents the length of the equal sides.