<h3>Answer: QS is 37 units long</h3>
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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
