Question

In the following picture, (QS) ̅ and (PR) ̅ are chords of the circle. Find the length of QS. (Remember to show formulas/equations and numbers in those formulas/equations to receive credit.)

Answer 1

Answer: QS is 37 units long

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Explanation:

PN = 24 and NR = 14 are pieces of the chord PR. The two pieces multiply to some value. At the same time, the other two pieces QN = 21 and NS = 2x-4 also multiply to that same value. In short, the chord pieces multiply to the same number. This is known as the intersecting chord theorem.

PN*NR = QN*NS

24*14 = 21*(2x-4)

336 = 42x-84

336+84 = 42x-84+84 .... add 84 to both sides

420 = 42x

42x = 420

42x/42 = 420/42 ...... divide both sides by 42

x = 10

We know that x = 10 so we can use it to find the length of NS

NS = 2x-4

NS = 2*10-4

NS = 20-4

NS = 16

Therefore,

QS = QN + NS

QS = 21 + 16

QS = 37

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note: as a check,

PN*NR = QN*NS

24*14 = 21*16

336 = 336

we get a true equation, so we have the proper values