NCERT Class 9 Solutions: Circles (Chapter 10) Exercise 10.6 – Part 5

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Understanding tangent, concentric and congruent circles

Understanding Tangent, Concentric and Congruent Circles

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Q-9 Two congruent circles intersect each other at point P and Q. Through P any line segment APB is drawn so that A, B lie on the two circles. Prove that QA=QB.

Solution:

Two Congruent Circles with Equal Radius

Given,

  • Two congruent circles intersecting each other at point P and Q.

  • PQ is a common chord of these circles.

Since PQ is common chord

  • Angle subtended by arcs of equal lengths on congruence circles must be equal, therefore,

  • Therefore in isosceles triangle AQB,

Q-10 In any triangle PQR, if the angle bisector of and perpendicular bisector of QR intersect, prove that they intersect on the circumcircle of the circumcircle of triangle PQR.

Solution:

Triangle PQR and Its Circumcircle with Center O

Given,

  • PQR is a triangle inscribed in a circle with centre at O, A is a point on the circle

  • PA is the internal bisector of and B is the mid-point of QR

Prove:

BA is the right bisector of QR

In , since both the arcs QA and AR subtend equal angle at the circumference (.

Therefore in triangle QAR,

  • (Given)

  • (common line)

Therefore, by SSS criterion of congruence,

Hence, (corresponding parts of congruent triangles)

Now,

  • ( )

Therefore, BA is the right bisector of QR.