What is \angle BAC equal to?
Direction: Consider the following for the two (02) items that follow :<br>A triangle ABC is inscribed in the circle x^{2}+y^{2}=100. B and C have coordinates (6, 8) and (-8, 6) respectively.
- A. \pi/2
- B. \pi/3 or 2\pi/3
- C. \pi/4 or 3\pi/4 ✓
- D. \pi/6 or 5\pi/6
Correct Answer: C. \pi/4 or 3\pi/4
Explanation
The center of the circle is O(0,0). The position vectors of B and C are \vec{OB} = \langle 6, 8 \rangle and \vec{OC} = \langle -8, 6 \rangle. The dot product \vec{OB} \cdot \vec{OC} = 6(-8) + 8(6) = 0, meaning the angle subtended by arc BC at the center is 90^{\circ} (\pi/2). The angle \angle BAC subtended by the arc at any point A on the circumference is half the center angle, which is 45^{\circ} (\pi/4). If A lies on the minor arc, the angle is 180^{\circ} - 45^{\circ} = 135^{\circ} (3\pi/4).
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