The bisector CN of triangle ABC divides side AB into segments AN = 6 and NB = 11. the tangent to the circumscribed

| July 25, 2021 | education

The bisector CN of triangle ABC divides side AB into segments AN = 6 and NB = 11. the tangent to the circumscribed circle of triangle ABC passing through point C intersects line AB at point D, find CD.

Consider triangles ВСD and АСD and prove that they are similar.

The angle D is common for the triangles. The angle ACD between the chord AC and the tangent CD is equal to half the degree measure of the arc AC, as is the inscribed angle ABC, then the triangles BCD and ACD are similar in two angles.

Then AD / CD = CD / BD = AC / BC.

By the property of the bisector of the triangle ABC, AC / BC = AM / BM = 6/11.

Then AD / CD = CD / BD = 6/11.

AD = 6 * CD / 11.

ВD = 11 * СD / 6.

BD = AB + AD = 18 + AD.

11 * CD / 6 = 17 + 6 * CD / 11.

CD * (11/6 – 6/11) = 17.

CD * (11 * 11 – 6 * 6) / 66 = 17.

CD * (85/66) = 17.

CD = 17 * 66/85 = 13.2 cm.

Answer: The length of the CD segment is 13.2 cm.



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