Tangent to a Circle Math Example 4

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Example 4

hard
A circle with center OO and radius r=6r = 6 is inscribed in angle ∠BAC\angle BAC (i.e., tangent to both rays ABAB and ACAC). The tangent points are DD on ABAB and EE on ACAC. If AD=9AD = 9, find AEAE, and then find the length AOAO.

Solution

  1. 1
    Step 1: By the Two-Tangent Theorem, AD=AE=9AD = AE = 9 (both tangent from the same external point AA).
  2. 2
    Step 2: To find AOAO, note that OD⊥ABOD \perp AB (radius to tangent point), so triangle ODAODA is right-angled at DD with OD=r=6OD = r = 6 and AD=9AD = 9.
  3. 3
    Step 3: Apply the Pythagorean theorem: AO2=AD2+OD2=81+36=117AO^2 = AD^2 + OD^2 = 81 + 36 = 117.
  4. 4
    Step 4: Simplify: AO=117=313≈10.82AO = \sqrt{117} = 3\sqrt{13} \approx 10.82.

Answer

AE=9AE = 9; AO=313≈10.82AO = 3\sqrt{13} \approx 10.82
The Two-Tangent Theorem immediately gives AE = AD = 9. Then using the right triangle formed by the radius, the tangent segment, and the line from the center to the external point yields AO = 3√13.

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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