Question

O is the circumcenter of Angle ABC and OAC =50° , then the value of Angle ABC is ...​

Answer 1

Answer:

In triangle ABC, o is the circumcenter and angle BAC=85 and angle BCA=75. What is angle OAC equal to?

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Given that in [math]\Delta ABC[/math] , [math]\angle BAC=85^\circ[/math] & [math]\angle BCA=75^\circ[/math]

[math]\therefore \angle ABC=180^\circ-\angle BAC-\angle BCA[/math]

[math]=180^\circ-85^\circ-75^\circ[/math]

[math]=20^\circ[/math]

Now, by property of chord of circle that the angle subtended by any chord at the center of circle is equal to double the angle subtended by the same chord at any point in the corresponding segment of circle,

[math]\therefore \angle AOC=2\angle ABC=2(20^\circ)=40^\circ[/math]

Now, in isosceles [math]\Delta AOC[/math] , [math]\angle OAC=\angle OCA[/math]

[math]\therefore \angle AOC+\angle OAC+\angle OCA=180^\circ[/math]

[math]40^\circ+2\angle OAC=180^\circ\quad (\because \ \angle OCA=\angle OAC)[/math]

[math]\angle OAC=\frac{180^\circ-40^\circ}{2}[/math]

[math]=70^\circ[/math]

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O is the circumcenter of triangle ABC . Therefore OA = OB = OC =radius of circumcircle.

In triangle ABC :-

Angle BAC+Angle ABC +Angle BCA=180°.

85°+AngleABC +75° =180°

Angle ABC =180°-160° = 20°.

In triangle OAC:-

OA =OC = radius of circumcircle.

so that angle OAC = angle OCA = x°(let).

Angle AOC = 2×Angle ABC = 2×20° = 40°.

Angle OAC + Angle OCA+ Angle AOC = 180°.

x`+ x° + 40° = 180°.

2x° = 180° - 40° = 140°.

x = 140°/2 = 70°.

Angle OAC = x° =70°