Tangent to a Circle Math Example 3

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Example 3

easy
Line ℓ\ell is tangent to circle OO at point TT. If the radius OT=5OT = 5 and a point AA on line ℓ\ell satisfies OA=13OA = 13, find ATAT.

Solution

  1. 1
    Step 1: Since ℓ\ell is tangent at TT, we have OT⊥ATOT \perp AT, so triangle OTAOTA is right-angled at TT. Use the Pythagorean theorem: AT2=OA2−OT2=169−25=144AT^2 = OA^2 - OT^2 = 169 - 25 = 144.
  2. 2
    Step 2: Therefore AT=12AT = 12.

Answer

AT=12AT = 12
A tangent is perpendicular to the radius at the point of tangency. Triangle OTA is a 5-12-13 right triangle (a well-known Pythagorean triple), so AT = 12.

About Tangent to a Circle

A line that touches a circle at exactly one point, called the point of tangency. At this point, the tangent line is perpendicular to the radius.

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Example 4 hard

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