The base of an isosceles triangle is 1/4 as long as the two equal sides. Write the area of the triangle as a function of the length of the base.

Let x be the length of the two equal sides.

x/4 is length of the base of triangle.

let height = h

x^2-(x^2/64)=h^2 (using pythagoras theorem, half of base = x/8)

(64x^2-x^2)/64 = h^2

h=(sqrt63*x)/8

Area = 1/2 * base* height

Area = 1/2 * (x/4) * (sqrt 63 * x)/8

Area=(x^2 / 32) * sqrt 63

Area = (x^2 / 32) * 3* sqrt 7

but x=4*b (where b is the base)

Area = ((4b)^2/32) * 3sqrt7

=16b^2/32 * 3sqrt7

= (b^2)/2 * 3sqrt7

=(3/2)sqrt7 * b^2

=3.96 b^2

To write the area of the triangle as a function of the length of the base, we need to find the relationship between the base and the other two sides.

Let's assume the length of each equal side is x. According to the given information, the base is 1/4 of the length of the two equal sides. Therefore, the length of the base is x/4.

To calculate the area of the triangle, we can use the formula:

Area = (1/2) * base * height

In an isosceles triangle, the height is the perpendicular distance from the base to the top vertex, and it bisects the base.

To find the height, we can use the Pythagorean theorem. Let's denote the height as h.

Using the Pythagorean theorem in the right-angled triangle formed by the base, height, and one of the equal sides gives us:

(x/4)^2 + h^2 = x^2

Simplifying this equation, we get:

x^2 / 16 + h^2 = x^2

Rearranging:

h^2 = x^2 - x^2 / 16

h^2 = (16x^2 - x^2) / 16

h^2 = 15x^2 / 16

Taking the square root of both sides:

h = √(15x^2 / 16) = (x√15) / 4

Now, we substitute the values of the base and height into the area formula:

Area = (1/2) * (x/4) * (x√15) / 4

Simplifying:

Area = (x^2√15) / 32

Therefore, the area of the isosceles triangle can be expressed as a function of the length of the base (x) as:

Area(x) = (x^2√15) / 32

To write the area of the triangle as a function of the length of the base, we first need to find the length of the equal sides.

Let's assume the length of each equal side is 's'. According to the given information, the length of the base is 1/4 times the length of the equal sides. Therefore, the length of the base (b) can be written as:

b = (1/4)s

Now, to find the area of the triangle, we can use the formula:

Area = (1/2) * Base * Height

In this case, the base is 'b', and we need to find the height.

To find the height of the isosceles triangle, we can use the Pythagorean Theorem. Since the triangle is isosceles, the height bisects the base and forms two congruent right triangles. Let's call the height 'h'.

By applying the Pythagorean Theorem to one of the right triangles, we have:
(s/2)^2 + h^2 = s^2
(s^2/4) + h^2 = s^2
h^2 = s^2 - (s^2/4)
h^2 = (3/4)s^2
h = sqrt((3/4)s^2)
h = (sqrt(3)/2)s

Now that we have the height 'h' in terms of the length of the equal sides 's', we can substitute it into the area formula:

Area = (1/2) * b * h
Area = (1/2) * (1/4)s * (sqrt(3)/2)s
Area = (1/8)s^2 * sqrt(3)

Therefore, the area of the isosceles triangle, A, can be expressed as a function of the length of the base, b:

A(b) = (1/8)b^2 * sqrt(3)
or
A(b) = (sqrt(3)/8)b^2