Q7 of 33 Page 51

In the picture, AB and CD are mutually perpendicular chords of the circle. Prove that arcs APC and BQD joined together would make half the circle.

Let O be the center of a circle


Join BC, OA, OB, OC and OD


Given: AB⊥ CD


In Δ ABK


∠ ABC + ∠ BCD + ∠ BKC = 180° (Angle sum property)


⇒ ∠ ABC + ∠ BCD + 90° = 180°


⇒ ∠ ABC + ∠ BCD = 90° …..(1)


∠AOC = 2∠ ABC (∵ angle at center twice angle in


segment) ...(2)


∠BOD = 2∠ BCD (∵ angle at center twice angle in


segment) ...(3)


Adding (2) and (3)


∠AOC + ∠BOD = 2∠ ABC + 2∠ BCD


∠AOC + ∠BOD = 2 (∠ ABC + ∠ BCD)


∠AOC + ∠BOD = 2 × 90° (from (1))


∠AOC + ∠BOD = 180°



Hence, (angle at center is half of 360° )


More from this chapter

All 33 →
5

In the picture, O is the centre of the circle and A, B, C are points on it. Prove that ∠ OAC + ∠ ABC = 90°.

 6

Draw a triangle of circumradius 3 centimetres and two of the angles and .

 8

In the picture, A, B, C, D are points on a circle centred at O. The line AC and BD are extended to meet at P. The line AD and BC intersect at Q. Prove that the angle which the small arc AB makes at O is the sum of the angles it makes at P and Q.

 1

Calculate the angles of the quadrilateral in the picture and also the angles between their diagonals: