Question

in figure A, B, C are three points on the circle with center O such that angle AOB= 90 Angle AOc = 110
find angle BAC​

Answer 1

Answer:

Angle BAC = 80°.

Step-by-step explanation:

angle AOB +angle AOC= 90°+110°= 200°= refex angle BOC.

And sum of all angles= 360°(complete circle).

angle AOB+angle AOC+ angle BOC = 360°.

angle BOC = 360°-200° = 160°.

Now, angle BAC= angle BOC/2.

angle BAC= 160°/2= 80°.

(This is because the angle in the segment is half the angle subtended at the center of the circle).

Answer 2

Given:

Angle AOB=90°

Angle AOC=110°

To find:

The measure of angle BAC

Solution:

The measure of angle BAC is 80°.

We can find the angle by following the given steps-

We know that O is the center of the circle and OA, OB, and OC are the radii of the circle.

So, OA, OB, and OC are equal in length.

In ∆ AOB, the sum of angle AOB, OAB, and OBA=180°

Angle AOB =90°

OA=OB (Radii of the circle)

So, angle OAB and angle OBA are equal. (Angles opposite to equal sides are equal)

Angle OAB +Angle OBA + Angle AOB=180°

Angle OAB +Angle OBA+90°=180°

Angle OAB +Angle OBA=90°

2× Angle OAB=90°

Angle OAB=45°

Angle OAB=Angle OBA=45°

Similarly, in ∆AOC,

Angle AOC=110°

OA=OC (Radii of the circle)

So, angle OAC and angle OCA are equal. (Angles opposite to equal sides are equal)

Angle AOC+ angle OAC+ angle OCA=180°

110°+angle OAC+ angle OCA=180°

Angle OAC+ angle OCA=70°

2×angle OAC=70°

Angle OAC=35°

Angle OAC=Angle OCA=35°

Now, to find the angle BAC, we need to add the angle OAB and angle OAC.

angle OAB + angle OAC= angle BAC

On putting the values,

Angle BAC= 45°+35°=80°

Therefore, the measure of angle BAC is 80°.