Question

ABC is a triangl BQ and CR are the bisectors of angles \(\angle\)ABC and \(\angle\)BCA respectively, Q and R are two points on the sides AC and AB respectively. The bisectors meet at a point O. If AQOR is a cyclic quadrilateral, find the measure of \(\angle\)BAC

• A. 90°
• B. 45°
• C. 30°
• D. 60°
• E. None of these

Answer 1

The Correct Option is A

The correct answer is option (A):

90°

Let's analyze the given problem step by step. We are given a triangle ABC, and BQ and CR are angle bisectors of angles ABC and BCA, respectively. The bisectors meet at a point O. Also, AQOR is a cyclic quadrilateral. We need to find the measure of angle BAC.

Since AQOR is a cyclic quadrilateral, the sum of opposite angles is 180 degrees. Therefore, angle QOR + angle BAC = 180 degrees.

In triangle BOC, angle OBC = angle ABC / 2 and angle OCB = angle BCA / 2 (because BQ and CR are angle bisectors). The sum of angles in triangle BOC is 180 degrees. Thus, angle BOC + angle OBC + angle OCB = 180 degrees.
So, angle BOC + (angle ABC / 2) + (angle BCA / 2) = 180 degrees.
This means angle BOC = 180 degrees - (angle ABC / 2) - (angle BCA / 2).
Also, angle QOR = angle BOC (vertically opposite angles).
Therefore, angle QOR = 180 degrees - (angle ABC / 2) - (angle BCA / 2).

Now, we know that the sum of angles in triangle ABC is 180 degrees. Therefore, angle BAC + angle ABC + angle BCA = 180 degrees.
We also know that angle QOR + angle BAC = 180 degrees.
Substituting the expression for angle QOR, we get:
[180 degrees - (angle ABC / 2) - (angle BCA / 2)] + angle BAC = 180 degrees.
Simplifying, we get angle BAC = (angle ABC / 2) + (angle BCA / 2).
Multiplying both sides by 2, we have:
2 * angle BAC = angle ABC + angle BCA.
We know that angle BAC + angle ABC + angle BCA = 180 degrees.
Substituting angle ABC + angle BCA = 2 * angle BAC, we get:
angle BAC + 2 * angle BAC = 180 degrees.
3 * angle BAC = 180 degrees.
angle BAC = 180 degrees / 3.
Therefore, angle BAC = 90 degrees.
The measure of angle BAC is 90 degrees.