On segment AB, point C is chosen so that AC = 14 and BC = 36. A circle is constructed with center A passing through C. Find the length of the tangent drawn from point B to this circle
Draw a segment AD, where D is the tangent point of the circle and the tangent.
AD is perpendicular to the tangent (by the tangent property), i.e. the angle between AD and tangent DB is 90 °.
Therefore, the triangle ABD is right-angled.
AD = AC = 14 (because these are the radii of a circle and, accordingly, are equal to each other).
By the Pythagorean theorem: AB2 = AD2 + BD2
(AC + BC) ^ 2 = AD ^ 2 + BD ^ 2
(14 + 36) ^ 2 = 14 ^ 2 + BD ^ 2
2500 = 196 + BD2
BD ^ 2 = 2304
BD = 48
Answer: the tangent length is 48
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