Question

is tangent to circle G at point F.

Circle G is shown. Line segment G F is a radius. Line segment F H is a tangent that intersects the circle at point F. A line is drawn from point H to point G and goes through a point on the circle. The length of F H is 35, and the length of the line segment from point H to the point on the circle is 25. The lengths of the radii are r.

What is the length of the radius, r?

10 units
12 units
20 units
24 units

Answer 1

Let GF be perpendicular to tangent FH, so triangle GHF is right with GF = r, FH = 35 and HG = sqrt(r^2 + 35^2).

The line HG meets the circle at a point P with HP = 25, and P is on the radius line, so PG = r. Thus HG = HP + PG = 25 + r.

So sqrt(r^2 + 35^2) = r + 25.
Square both sides: r^2 + 1225 = r^2 + 50r + 625 → 1225 − 625 = 50r → 600 = 50r → r = 12.

Answer: 12 units.