The quadrilateral ABCD is inscribed in a circle. The diagonal AC is the bisector of the angle
| September 23, 2021 | education
The quadrilateral ABCD is inscribed in a circle. The diagonal AC is the bisector of the angle BAD and intersects with the diagonal BD at point K. Find KC, if BC = 4, and AK = 6.
Consider triangles BKC and ABC. The angle C is common for the triangles. Angle КBC = DBC. The angle DBC rests on the chord DC and the angle DAC rests on the chord BC, then the angle DВC = DAC = KBC. Since BAC = DAC by condition, the angle BAC = KBC, and the triangles are similar in two angles.
Let the segment KC = X cm, then AC = (X + 6) cm.
Then BC / AC = KC / BC.
4 / (X + 6) = X / 4.
16 = X2 + 6 * X.
X^2 + 6 * X – 16 = 0.
Having solved the quadratic equation, we get X = 2.
COP = 2 cm.
Answer: The KM segment is 2 cm.
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