SOLUTION: AB is the diameter of a circle. AD and BC are tangents to the circle with AD=9 cm and BC=16 cm. If AC and BD intersect at a point on the circle, then what is the length of AB in cm

Algebra ->  Circles -> SOLUTION: AB is the diameter of a circle. AD and BC are tangents to the circle with AD=9 cm and BC=16 cm. If AC and BD intersect at a point on the circle, then what is the length of AB in cm      Log On


   

Question 1133555: AB is the diameter of a circle. AD and BC are tangents to the circle with AD=9 cm and BC=16 cm. If AC and BD intersect at a point on the circle, then what is the length of AB in cm?
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


If AD and BC are tangents to a circle at the endpoints of diameter AB, and if AC and BD intersect at a point on the circle, then AD and BC are perpendicular to diameter AB and are on the same side of AB.

Let E be the point of intersection of AC and BD; E is on the circle. Let F be on AB with EF perpendicular to AB.

Using similar triangles (or other methods) it can be determined that EF = (AD*BC)/(AD+BC) = 144/25.

Also using similar triangles, it can be determined that AF:FB = 9:16.

So let AF=9x anf FB=16x; then once again similar triangles tell us that

AF%2FEF=EF%2FFB
9x%2F%28144%2F25%29=%28144%2F25%29%2F16x
144x%5E2+=+%28144%2F25%29%5E2
x%5E2+=+144%2F625
x+=+12%2F25

Then the length we are looking for, AB, is AF+FB = 25x = 12.