Question

Two circles intersect in A and B. AC and AD are the diameters of the circles.
Prove that C, B and D are collinear.

Answer 1

Join CD, BD and AB.

Since, The angle in a semicircle is a right angle.

Therefore, AC is a diameter of the circle with centre at O.



We know  that, angles in a semi-circle is 90°
∴ ∠ABC = 90°    ...(1)


Also, AD is a diameter of the circle with centre O’

∠ABD = 90°  ……….....(2)


On Adding eq. (1) and (2),


∠ABC + ∠ABD = 90° + 90°


∠ABC + ∠ABD = 180°


∠CBD = 180°


CBD is a straight line.


Hence,C, B & D are collinear

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Hope this will help you... .

Answer 2

GIVEN:

• Two circles with centres O and O' respectively, intersect each other at points A and B.
• AC and AD are the diameters of the circles.

TO PROVE:

• Prove that the points C, B and D are collinear.

CONSTRUCTION:

Join AB

PROOF:

AC is diameter of circle with centre O

and,

Arc ABC is a semicircle

【∴ Angle in a semicircle is 90°】

So,

∠ ABC = 90°

Similarly,

AD is the diameter of circle with centre O'

Arc ABD is semicircle

So,

∠ ABD = 90°

Now,

$\sf{\implies \angle ABD + \angle ABC = 180^°}$

$\sf{\implies 90^° + 90^° = 180^°}$

∴ Since, the sum of angles is 180°, they form a linear pair.

Linear pair can only happen on a line.

Therefore,

➨ DBC is a straight line.

So,

D, B and C are collinear.

Hence proved