The height of an isosceles triangle is 4r. The radius of an inscribed circle is r. What is the area of the triangle?
the info of about the circle is irrelevent. Since the triangel is isosceles than that means another side of it is 4r. Find the missing side then use the area of a triangle formula!
The 'height' of a (non-right-angled) triangle is not the length of one side.
Even if you were given one side length of an isosceles triangle, you would *not* necessarily know the length of another side, as the side given could be the odd one out.
Area of a circle:
A = (1/2) * base * height
Given that height = 4r, we already know that:
A = (1/2) * base * 4r
So you just need to find the base of the triangle in terms of r.
Draw a circle inscribed within an isosceles triangle.
Draw a line from the middle of the triangle base going through the center of the circle to the vertex of the triangle. That line marks the height of the triangle. Note that the line divides an isosceles triangle's base exactly in half.
From the center of the circle, draw a line to one of the sides of the triangle such that the line is perpendicular to the side.
Call the total base '2b' the height 'h', the radius of the circle 'r' and the length of the side of the triangle 's', you should see that:
A = (1/2) * (2b) * 4r
= 4br
and
h^2 + b^2 = s^2
and
(h-r)^2 = r^2 + (s - b)^2
Now remember you were given that:
h = 4r
So:
Express s in terms of r and b.
Then express b in terms of r.
Then plug that into your original equation for area.
Excellent question.
To find the area of the triangle, we need to determine the length of its base.
Since the triangle is isosceles, it has two equal sides. Let's denote the length of each of these sides as "a".
We can draw a line segment from the top vertex of the triangle to the midpoint of the base, creating two congruent right triangles. In each right triangle, the height is 4r, and the hypotenuse is a.
Using the Pythagorean theorem, we can determine the length of the base (2r) as follows:
a^2 = (2r)^2 + (4r)^2
a^2 = 4r^2 + 16r^2
a^2 = 20r^2
a = √(20r^2)
a = 2√5r
Now that we know the length of the base (2√5r) and the height (4r), we can calculate the area of the triangle using the formula:
Area = (base * height) / 2
Area = (2√5r * 4r) / 2
Area = (8√5r^2) / 2
Area = 4√5r^2
Therefore, the area of the isosceles triangle is 4√5r^2.