Find the area of a circle within a right angled triangle.

Question

Hi, I’ve found this question on my school quiz but am really confused in how i should approach it and find the answer. In other words, I’m completely lost.

A circle of unknown radius fits perfectly in a triangle. The triangle has sides of 3,4 and 5 units. Find the area of the circle given that the circle touches all three sides.

Answer 1

Answer 2

If you draw the radii of the circle that intersect each side, we see that we have created isosceles kites and a square with sides = to radius of circle, r.

We can label the

base = r + x = 4 square base + a length we call x

perpendicular = r + y = 3 square side + a length we call y

hypotenuse = x + y = 5 because the kites are isosceles (or rays from an external angle tangent to circle)

We can solve for r by taking the 3rd equation and subtracting the first and the second equation from it:

-2r = 5 - 4 - 3 or r = 1 You can now find the area of the circle.

Not that generally for right angles r = (1/2) (b + p - h)

Hope that helps!

Answer 3

The number 3,4 and 5 remind you of pythagoras theorem (9+16=25). So draw a right angled triangle and inscribe a circle in it to touch the 3 sides. I want to upload an image to make explaining easier. From the center of the circle drop perpendiculars on the side of length 3, 4 and 5. Length of each perpendicular is the radius r. So the remaining length of two sides of the triangle is 3-r and 4-r. Draw a line joining center and vertex on the right. This splits it into 2 identical right triangles, with base = 4-r for each. The upper triangle's side of length 4-r lies on the side that is 5 units. So the remaining side length = 5-(4-r). Doing the same thing to the top vertex, 5-(4-r) = 3-r. This gives r = 1. Therefore area of circle = ?