The intersection points of the chords AB and CD divides the angle D into segments CN = 4cm, and ND = 6cm.

| September 30, 2021 | education

The intersection points of the chords AB and CD divides the angle D into segments CN = 4cm, and ND = 6cm. Into what segments does point N divide the chord AB = 11cm?

Let the length of the segment NB = X cm, then the length of the segment NA = AB – X = 11 – X cm.

By the property of intersecting chords of a circle, the product of the lengths of the segments formed at the intersection of one chord is equal to the product of the lengths of the segments of the other chord.

NА * ОВ = NС * NД.

(11 – X) * X = 4 * 6.

– X ^ 2 + 11 * X = 24.

X ^ 2 – 11 * X + 24 = 0.

Let’s solve the quadratic equation.

D = b ^ 2 – 4 * a * c = (-11) ^ 2 – 4 * 1 * 24 = 121 – 96 = 25.

X1 = (11 – √25) / (2 * 1) = (11 – 5) / 2 = 6/2 = 3.

X2 = (11 + √25) / (2 * 1) = (11 + 5) / 2 = 16/2 = 8.

NB = 3 cm.

NA = 11 – 3 = 8 cm.

Answer: Point N divides chord AB into 3 cm and 8 cm segments.



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