A circle is inscribed in square ABCD with side a, which touches side CD at point E.

| April 28, 2021 | education

A circle is inscribed in square ABCD with side a, which touches side CD at point E. Find the length of the chord connecting the points at which the circle meets the line AE.

Segment AE intersects the circle at point K.

Triangle ADE is rectangular, in which leg AD = a cm, leg DE = a / 2 cm.

Then, by the Pythagorean theorem, AE ^ 2 = AD ^ 2 + DE ^ 2 = a ^ 2 + a ^ 2/4 = 5 * a2 / 4.

AE = a * √5 / 2 cm.

Segment AP is tangent to the circle, AE is secant.

Then, by the theorem on the tangent and secant drawn from one point: AP ^ 2 = AK * AE.

AK = AP ^ 2 / AE = (a / 2) ^ 2 / (a * √5 / 2) = (a ^ 2/4) / (a * √5 / 2) = a / 2 * √5 = a * √5 / 10cm.

KE = AE – AK = a * √5 / 2 – a * √5 / 10 = 4 * a * √5 / 10 = 2 * a * √5 / 5 cm.

Answer: The length of the chord is 2 * a * √5 / 5 cm.



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