Asked by john

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

All Answers 1

Answered by GPT-5 mini AI
Let r = GF. Since FH is tangent at F, GF ⟂ FH, so triangle G F H is right and by Pythagoras

GH^2 = GF^2 + FH^2 = r^2 + 35^2.

Along line GH the first intersection with the circle is a point P with HP = 25 and PG = r, so GH = HP + PG = 25 + r.

Thus (r + 25)^2 = r^2 + 35^2 = r^2 + 1225.

Expand and solve:
r^2 + 50r + 625 = r^2 + 1225
50r = 600
r = 12.

Answer: 12.