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In the given figure, two circles intersect at M and N. ABC is a triangle. State whether the statement is true or false:

A,L,M, and K are concyclic points.

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True

False

answer is A.

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Detailed Solution

It is given that; two circles intersect at M and N and ABC is a triangle.
Question Image

We need to find if A,L,M, and K are concyclic points is true or false.
We know that a cyclic quadrilateral's opposing angles are supplementary.
That is, the total of the opposite angles equals

180 °

.
The exterior angle of a cyclic quadrilateral equals the interior opposite angle.
In the given figure, join MN,LM,KM .
Question Image

Observe that LMNB lie on the same circle.
So, LMNB is a cyclic quadrilateral.
KMNC is also a cyclic quadrilateral.

∠ 1 + ∠ 2 = 180 °

(Property of cyclic quad.) ...(i)
and

∠ 3 + ∠ 4 = 180 °

(Property of cyclic quad.) ...(ii)

∠ 1 = ∠ 4

(exterior angle property of cyclic quadrilateral) ...(iii)
and

∠ 3 = ∠ 2

(exterior angle property of cyclic quadrilateral) ...(iv)
We can observe that, BNC is a line segment and so,

∠ 2 + ∠ 4 = 180 °

.
Solve equations (iii) and (iv).

⇒ ∠ 1 = ∠ 3 = ∠ 2 = ∠ 4 = 90 ° … (v)

By the rule of linear pair angles,

⇒ ∠ 1 + ∠ M L A = 180 ° ⇒ ∠ 3 + ∠ M K A = 180 °

Now, use equation (v) and solve further.

⇒ ∠ M L A = ∠ M K A = 90 °

The sum of supplementary angles is

180 °

∠ M L A + ∠ M K A = 180 °

Therefore, LMKA is a cyclic quadrilateral.
So, A,L,M, and

K

are concyclic points.
Given statement is true.
The correct option is 1).

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