Question

Draw a circle of radius 3.4 cm and centre O. Draw any radius OP. Make an anglePOQ = 60 degree with the point Q on the circle. Join PQ. Find the length of chord PQ and also the distance of chord from the centre.

Answer 1

Draw a circle of radius $3.4 \, \text{cm}$ and centre O. Draw any radius OP. Make an angle $\text{POQ} = 60$ degree with the point Q on the circle. Join PQ. Find the length of chord PQ and also the distance of chord from the centre.

Solution:

In $\Delta POQ$ :

$OQ = OP = 3.4cm$ as the radiuses of a circle.

If two sides of a triangle are equal, then the angles opposite those sides are equal, so

$\angle OQP = \angle OPQ$

The sum of angles of the triangle is $180{}^{\circ}$ :

$\angle OQP + \angle OPQ + \angle POQ = 180{}^{\circ}$

$\angle POQ = 60{}^{\circ}$ - given, so

$2\angle OQP + 60{}^{\circ} = 180{}^{\circ}$

$\angle OQP = \angle OPQ = \angle POQ = 60{}^{\circ}$

If two angles of a triangle are equal, then the sides opposite those angles are equal, so

$PQ = OQ = OP = 3.4cm$

The distance of chord from the centre is the length of the perpendicular $OH$ to $PQ$ .

Since $\Delta POH$ is the right triangle, then

$OH = OP\sin \angle OPQ = 3.4\sin 60{}^{\circ} = 1.7\sqrt{3} = 2.94cm$

Answer: $PQ = 3.4cm$ , the distance of chord from the centre is $2.94cm$ .