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High School Mathematics - 2
8.6 Circles - Theorems

Statement: If two arcs are congruent, then their chords are equal.
Given: Arc ASB = Arc CTD
To prove: AB = CD

Proof: OA = OlC, OB = OlD (radii)
Angle AOB = Angle COlD (given that the arcs are congruent)
Hence by SAS Postulate on congruence AOB is congruent to COlD
Hence AB = CD

Statement: There is only one circle passing through three non collinear points.

Example: In the given figure prove that AB = CD

Solution: Arc BAD congruent Arc CDA
Hence Arc BAD - Arc AD = Arc CAD - Arc AD
Arc BA is congruent Arc CD
Hence AB = CD


Directions: Solve the following.
Q 1: BC is a chord of circle with centre O. A is a point on the major arc BC as shown, prove that angBAC + angOBC = 90.

Answer:

Q 2: Find the missing angle.

55o
30o
35o

Q 3: Two diameters of a circle intersect each other at right angles. Prove that the quadrilateral formed by joining their end points is a square.
Answer:

Q 4: Two circles intersect each other at points A and B, if AP and AQ be the respective diameters of the circle prove that PBQ is a line.
Answer:

Q 5: AC and BD are chords of a circle that bisect each other. Prove that AC and BD are diameters.
Answer:

Q 6: D is the midpoint of side BC of isosceles triangle ABC with AB = AC. Prove that the circle drawn witrh either of the equal sides as diameter pass through D.
Answer:

Q 7: In a circle containing the equal chords AB and CD which are on the opposite side of the centre, E. Prove that BE = DE and AE = CE where E is the point of intersection of AD and BC.
Answer:

Q 8: Find angle x

55o
35o
90o

Question 9: This question is available to subscribers only!

Question 10: This question is available to subscribers only!


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