Through point K, located at a distance a from the center of a circle with a radius

Through point K, located at a distance a from the center of a circle with a radius of r = 13 cm, a chord AB is drawn. Find its length if: a = 8, AK-BK = 8

From point O, the center of the circle, draw a perpendicular OH to the chord AB, then AH = BH.

AH = AK – KH.

BH = BK + KH.

Let us equate both equalities, since AH = BH.

AK – KH = HK + KH.

AK – HK = 2 * KH.

2 * KH = 8.

KH = 4 cm.

In a right-angled triangle OHK, according to the Pythagorean theorem, OH ^ 2 = OK ^ 2 – KH ^ 2 = a ^ 2 – 16.

The triangle OBH is rectangular, then BH ^ 2 = OB ^ 2 – OH ^ 2 = 169 – (a ^ 2 – 16) = 185 – a2.

BH = √ (185 – a2).

Then AB = 2 * √ (185 – a2) see.

Answer: The length of the chord is 2 * √ (185 – a2) cm.