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High School Mathematics - 28.4 Theorems - Circles

 Theorem: Angle is a semi circle is a right angle. Given : Angle AOB is in semi circle. To prove: Angle ACB = 90o Proof: Angle ACB = 1/2 angle AOB (angle subtended at the centre is twice angle at circumference) Angle AOB = 180o (straight line) Hence ACB = 90o Theorem: Angles in the same segment of a circle are equal. Directions: Solve the following
 Q 1: ABC is a triangle and P is a point on BC such that AB = BP. If AQ is produced to meet the circumcircle of triangle ABC at Q. Prove that CP = CQAnswer: Q 2: If diagonals AC and BD of a quadrilateral intersect at M. Prove that a line drawn through M to bisect any side of the quadrilateral is perpendicular to the opposite side.Answer: Q 3: In the figure, PAQ and RAS are straight lines. Show that angPXR = angQYS. Answer: Q 4: find x.90 degrees55 degrees110 degrees Q 5: Find the angle x.90 degrees70 degrees17.5 degrees Q 6: In the figure, CAD and CBE are straight lines. If CA is the diameter of the circle ABC, find angle ADE180 degrees45 degrees90 degrees Q 7: Two circles intersect each other at points A and B. If AP and BQ be the respective diameters, show that PBQ is a straight line.Answer: Q 8: Angles in a semicircle is always ____.90 degrees270 degrees180 degrees Question 9: This question is available to subscribers only! Question 10: This question is available to subscribers only!

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