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Question

Two circles each of which passes through the centre of the other intersect points M and N. A line from M intersects the circle at K and L as shown in the figure. If KL = 6 compute the area of ∆KLN.

I was attempting a Practice Paper and stumbled on this Question.

I think the tangent secant theorem must used somewhere which states that.

$Tangent^2=Outer Secant×Whole Secant$

But I don't know where to apply it so I am clueless.

I don't know where to start from.

Any help will be appreciated

Crocogator
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1 Answers1

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Realize that $\angle NKM= 120^o$ and that $\angle KLN= 60^o$, which leaves $\Delta KLN$ an equilateral triangle. The answer is therefore $9\sqrt{3}$.

Community
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Larry
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  • Here is link for the basic angle theorems in a circle: https://mathbitsnotebook.com/Geometry/Circles/CRAngles.html – Larry Oct 16 '19 at 18:27
  • How did you find the aswer to be 9√3? – Crocogator Oct 17 '19 at 02:52
  • How did you find angle NKM to be 120°? I've gone through the links but none of the theorems help me find the angle. – Crocogator Oct 17 '19 at 03:01
  • I added another image as a hint. You already know that $\angle KLN= 60^o$, so according to the theorem I just added, $\angle KLN = 180^o - \angle KLN= 180^o- 60^o=120^o$. And the area of an equilateral triangle can be calculated if you know one side of the triangle. – Larry Oct 17 '19 at 11:27
  • $\angle NOM=\angle NKM$ since these two angles are on the same intercepted arc. – Larry Oct 17 '19 at 11:31