State true or false.

If two equal circles intersect in P and Q , with a straight line through P which meets the circle at points A and B, then QA=QB.

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True

False

answer is A.

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Detailed Solution

It is given that that two equal circles intersect in P and Q, with a straight line through P which meets the circle at points A and B.
We have to check if QA=QB.
We know that if two circles have an equal radius, then they are congruent to each other and corresponding arcs of those circles are congruent when two chords of the circles are equal.
Question Image

It is clear from the figure that PQ is the common chord for both circles.
Since PQ is a common chord, so their corresponding arcs are equal.
Therefore, arc PCQ= arc PDQ
Also, the angles under those congruent arcs are the same.

∴ ∠ P A Q = ∠ P B Q

Using the property of isosceles triangle to prove that QA is equal to QB.
The property says that when the base angles of a triangle are equal, then the triangle is called an isosceles triangle.
So, QA=QB
Thus, the statement is true.
Therefore, option 1 is correct.

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