MilesMaths : Five Triangles - Circle Area Ratio Solution

This is the solution to the problem:

To solve the problem above, you have to find the radius of each of the circles. To find the radius of the larger circle inside the triangle...

DG=EG=GF. It is the radius of the circle, so lets just say that they equal X.

AC=16 as shown above.

Therefore AD=16-X

AF=AD

AD=16-X

CB=12

Therefore EB=12-X

EB=BD

BD=12-X

BD+AD=AB=20

12-X+16-X=20

28-2X=20

2X=8

X=4

The radius of the larger circle is 4 units. So the area of it is 4x4xpi= 16pi

To find the radius of the smaller circle:

To find the radius of the smaller circle, tangent to the triangle, you have to form a sector.

The radius of the sector is 10, as the diameter (BC) is 20.

AB=BC=10 (The radius of the whole semi-circle)

AC=16, therefore EC=8

Sides EC and BC can form the sides of a right angled triangle, along with BE.

BE^2+EC^2=BC^2

BE^2+64=100

BE^2=36

BE=6

BF=10 as it is the radius on the whole sector.

BF-BE=EF (diameter of the small circle)

10-6=EF

EF=4

Therefore, if the diameter is 4, the radius must be 2.

If the radius is 2, Area=4pi

The question asks for the ratio of the two circles areas, not just the area of each.

Ratio = Area of the larger circle/Area of the smaller circle

= 16pi/4pi

So the larger circle inside the triangle, is 4 times bigger than the circle tangent to both AC and the semi-circle.

MilesMaths