Question

(1) A triangle ABC is inscribed in a circle with centre O. If ∠OBC=40°, then the measure of∠BAC is
(a)50° (b)40° (c)45° (d)55°
(2) O is mid point of a line segment AB. Two circles are drawn with AO and BO as diameters. If a straight line through O cuts the circles at P and R respectively, then justify that OP= OR​

Answer 1

Solution :

1.

A triangle ABC is inscribed in a circle with centre O.

Angle OBC is given as 40° .

We have to find the measure of angle BAC .

See the attachment .

Here , AB = AC ( Equal chords) .

OB = OC .

∆ OBC is an isosceles triangle .

Angle OBC = Angle OCB = 40° .

Angle BOC = 180 - 2×40 = 180 - 80 = 100°

Now

Angle BAC = ½ Angle BOC = 100/2 = 50°

Answer - (a) 50°

2.

O is the mid point of a line segment AB.

Two circles are drawn with AO and BO as diameters.

If a straight line through O cuts the circles at P and R respectively, then justify that OP= OR

The diagram is in the attachément .

Note that the line through O can cut through any any point , not necessarily directly perpendicular to the centre .

To show that OP = OR , the first step is to drop a perpendicular from P and R to the line AB .

Now if we can show these triangles formed congruent then by CPCT , it can be proved .

Consider the triangles ∆ OCP and ∆ ODR

Angle COP = Angle DOR ( Vertically opposite angles )

CO = OD ( As O is the midpoint of the line segment and these bisect it further )

CP = DR = Radius of the circles .

So , the triangles are congruent by AS S criteria .

By CPCT , OP = OR

Hence Shown .

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Answer 2

1]

Given :-

A triangle ABC is inscribed in a circle with centre O. If ∠OBC=40°,

To Find :-

Measure  of∠BAC is

Solution :-

Centre is O

According to the property of triangle

OBC = OCB

40 + 40 = 80

Sum of all angle in triangle = 180

Angle BOC = 180 - 80 = 100

Angle BAC = Angle BOC/2

Angle BAC = 100/2

Angle BAC = 50

2]

Given :-

Two  circles are drawn with AO and BO as diameters. If a straight line through O cuts the circles at P and R respectively,

To Prove :-

OP = OR

Solution :-

Two circle are AO and BO

Since, it is a straight line.

By using square set make a perpendicular to P and R

Accoridng to CPCT theorem the OP = OR