The chord perpendicular to the diameter divides it into segments, the difference of which is 7 cm.

| January 1, 2021 | education

The chord perpendicular to the diameter divides it into segments, the difference of which is 7 cm. Find the radius of the circle if the chord length is 24 cm.

Let the segment AK, which cuts off the chord from the diameter, is equal to X cm, then the segment BA, by condition, is equal to (X + 7) cm.
Segment SK = DC, since the chord is perpendicular to the diameter and is divided by it into equal segments
By the property of chords intersecting at one point, 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.
AK * DK = CK * DK.
X * (X + 7) = 12 * 12 = 144.
X ^ 2 + 7 * X – 144 = 0.
Let’s solve the quadratic equation.
D = b ^ 2 – 4 * a * c = 7 ^ 2 – 4 * 1 * (-144) = 49 + 576 = 625.
X1 = (-7 – √625) / (2 * 1) = (-7 – 25) / 2 = -32 / 2 = -16. (Doesn’t fit because <0).
X2 = (-7 + √625) / (2 * 1) = (-7 + 25) / 2 = 18/2 = 9.
AK = 9 cm, then BK = 9 + 7 = 16 cm.
Diameter AB = AK + BK = 25 cm, radius AO = 25/2 = 12.5 cm.
Answer: The radius of the circle is 12.5 cm.



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