PQRS is a cyclic quadrilateral with PQ = 11 RS = 19. M and N are points on PQ and RS respectively such that PM = 6, SN = 7, and MN = 27. The length of the line segment formed when MN is extended from both side until it reaches the circle is ?
Answers
Given :- (from image.)
- PQRS is a cyclic quadrilateral.
- PQ = 11
- RS = 19.
- M and N are points on PQ and RS respectively.
- PM = 6
- SN = 7
- MN = 27.
- KM = Let y .
- NL = Let x .
Solution :-
we know that,
- When two chords intersect each other inside a circle, the products of their segments are equal.
So,
→ KM * ML = PM * MQ
→ y(27 + x) = 6 * 5
→ 27y + yx = 30 ------------ Eqn.(1)
and,
→ KN * NL = RN * NS
→ (y + 27)x = 12 * 7
→ 27x + yx = 84 ----------- Eqn.(2)
Subtracting Eqn.(1) from Eqn.(2) ,
→ (27x + yx) - (27y + yx) = 84 - 30
→ 27x - 27y + yx - yx = 54
→ 27(x - y) = 54
→ (x - y) = 2
→ x = (y + 2) ------------ Eqn.(3)
Putting value of x in Eqn.(1),
→ 27y + y(y + 2) = 30
→ 27y + y² + 2y = 30
→ y² + 29y - 30 = 0
→ y² + 30y - y - 30 = 0
→ y(y + 30) - 1(y + 30) = 0
→ (y + 30)(y - 1) = 0
→ y = 1 or (-30) . {since Negative value of y is not Possible.}
Therefore,
→ y = 1 .
Putting value of y in Eqn.(3) ,
→ x = y + 2 = 1 + 2 = 3 .
Hence,
→ KL = y + 27 + x
→ KL = 1 + 27 + 3
→ KL = 31 (Ans.)
∴ The length of the line segment formed when MN is extended from both side until it reaches the circle is 31 .
{ Excellent Question. }
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