Question

In ΔABC, PQ is perpendicular bisector of side AB and PR is perpendicular bisector of side BC.

Statement (1): Perpendicular bisector of side AC will pass through point P.

Statement (2): Perpendicular bisectors of the sides of a triangle are concurrent.

Options

• Both the statements are true.

• Both the statements are false.

• Statement 1 is true, and statement 2 is false.

• Statement 1 is false, and statement 2 is true.

Answer 1

Both the statements are true.

Explanation:

Since PQ is the perpendicular bisector of AB, any point on PQ is equidistant from A and B.

Similarly, since PR is the perpendicular bisector of BC, any point on PR is equidistant from B and C.

Therefore, point P, which lies on both PQ and PR, is equidistant from A, B and C.

This means that P is the center of a circle passing through A, B and C.

The perpendicular bisectors for sides AB, BC, and CA always meet at a single point called the circumcenter of ∆ ABC that is equidistant from all three vertices.

Thus, we can say that,

Perpendicular bisectors of sides of a triangle are concurrent.

So, statement 2 is true.

Consequently, the perpendicular bisector of AC must also pass through P, as P is equidistant from A and C.

So, statement 1 is true.

∴ Both the statements are true.