Question

Triangle ABC is isosceles with AB=CB. Circle M is inscribed in Triangle ABC such that it is tangent at points D, E, and F. If the length of BF is twice the length of CF and the perimeter of Triangle ABC is 32 inches, then determine the length of side BC in inches.

Answer 1

The answer is "12 inches".

Explanation:

Please find the image file of the graph.

We know
`AE = CF \ and \ EB = FB` so because the triangle is isosceles.

`EB = 2x, FB = 2x,\ and \ CF = x` when
`AE` is called by
`x`.

We know that
`AE = AD = x` since E and D are tangent points to a circle.

They have
`CD = CF = x` since D and F are tangent points only to circle.

Thus, if the perimeter is 32, the following is the result:

`\to AE + EB + BF + FC + CD + DA = 32\\\\\to x + 2x + 2x + x + x + x = 32\\\\\to 8x = 32\\\\\to x = 4`

Therefore the length of
`BC = BF + FC = 2x + x = 3x = 12\ inches`