You visited us 1 times! Enjoying our articles? Unlock Full Access!

Sum of Opposite Angles of a Cyclic Quadrilateral

In the follow...

Question

In the following figure, AB is the diameter of a circle with centre O and CD is the chord with length euqal to radius OA.

If AC produced and BD produced meet at point P; show that : $∠ A P B = 60^{\circ} .$

Open in App

Solution

Given: $AB$ is diameter, $CD = OA = Radius$

In $\triangle CDO, CD = OD = OC$

$\Rightarrow \triangle CDO$ is equilateral

Therefore $\angle OCD = \angle OCD = \angle COD = 60^o$

In $\triangle AOC$

$OA = OC$ (radius of the same circle)

Therefore $\angle OCA = \angle CAO$

Similarly in $\triangle BOD$

$OD = OB$ (radius of the same circle)

Therefore $\angle ODB = \angle BDO$

Since $ABCD$ is cyclic quadrilateral

Therefore $\angle ACD + \angle OBD = 180^o$ (opposite angles of a cyclic quadrilateral are supplementary)

$60+ \angle ACO + \angle OBD = 180^o$

$\Rightarrow \angle ACO + \angle OBD = 120^o$

In $\triangle APB$

$\angle APB + \angle PBA + \angle BAP = 180^o$

$\angle APB + 120 = 180^o$

$\Rightarrow \angle APB = 60^o$ . Hence proved.

$\\$

Suggest Corrections

Similar questions

In a circle, with centre O, a cyclic quadrilateral ABCD is drawn with AB as a diameter of the circle and CD equal to radius of the circle. If AD and BC produced meet at point P; show that $∠$ APB = $60^{o}$ .

ABCD is a cyclic quadrilateral of a circle with centre O such that AB is a diameter of this circle and the length of the chord CD is equal to the radius of the circle. If AD and BC produced meet at P, show that $A P B = 60^{\circ} .$

Q. In the given figure,

A B

is the diameter of a circle

C (O, r)

. chord

C D

is equal to radius

O C

. If

A C a n d B D

intersect at

P

, find

∠ A P B

AB is the diameter and AC is a chord of a circle with centre O such that angle BAC = $30^{o}$ . The tangent to the circle at C intersects AB produced in D. Show that : BC = BD.

Q. In the figure,

A B

is the diameter of a circle with centre

O

. Chord

D C

is equal to radius of the circle.

P

is an external point. Find the measure of

∠ A P B

Join BYJU'S Learning Program

In the following figure, AB is the diameter of a circle with centre O and CD is the chord with length euqal to radius OA. If AC produced and BD produced meet at point P; show that : ∠APB=60∘.

Given: is diameter, In is equilateral Therefore In (radius of the same circle) Therefore Similarly in (radius of the same circle) Therefore Since is cyclic quadrilateral Therefore (opposite angles of a cyclic quadrilateral are supplementary) In . Hence proved.

In the following figure, AB is the diameter of a circle with centre O and CD is the chord with length euqal to radius OA.

If AC produced and BD produced meet at point P; show that : $∠ A P B = 60^{\circ} .$