In a ΔABC, the bisectors of ∠B and ∠C meet at point O, inside the triangle. If ∠BOC = 122°, then the measure of ∠A is:
64°
Understanding Angle Bisectors in a Triangle
The problem asks us to find the measure of angle A in a triangle ABC, given that the angle bisectors of ∠B and ∠C meet at a point O inside the triangle, and the angle ∠BOC is 122°.
Given Information:
- In ΔABC, the angle bisectors of ∠B and ∠C meet at O.
- ∠BOC = 122°.
Concept of Angle Bisectors and Incenter
The point where the angle bisectors of the angles of a triangle meet is called the incenter. This point is equidistant from the sides of the triangle. There is a specific relationship between the angle formed by the bisectors of two angles (∠BOC in this case) and the third angle (∠A) of the triangle.
Formula Relating ∠BOC and ∠A
The relationship between the angle formed by the bisectors of two angles (say ∠B and ∠C) at their intersection point O and the third angle (∠A) of the triangle is given by the formula:
\(\angle \text{BOC} = 90^\circ + \frac{\angle \text{A}}{2}\)
Step-by-Step Calculation of ∠A
We are given ∠BOC = 122°. We can substitute this value into the formula and solve for ∠A.
- Write down the formula:
\(122^\circ = 90^\circ + \frac{\angle \text{A}}{2}\)
- Subtract 90° from both sides of the equation:
\(122^\circ - 90^\circ = \frac{\angle \text{A}}{2}\)
- Calculate the difference:
\(32^\circ = \frac{\angle \text{A}}{2}\)
- Multiply both sides by 2 to find ∠A:
\(\angle \text{A} = 32^\circ \times 2\)
- Calculate the final value:
\(\angle \text{A} = 64^\circ\)
Thus, the measure of angle A is 64°.
Verification
Let's check if our answer is consistent using the formula:
\(90^\circ + \frac{64^\circ}{2} = 90^\circ + 32^\circ = 122^\circ\)
This matches the given value of ∠BOC, so our calculation is correct.
Conclusion
Based on the properties of angle bisectors meeting at the incenter, and using the formula \(\angle \text{BOC} = 90^\circ + \frac{\angle \text{A}}{2}\), we found that the measure of ∠A is 64°.
Revision Table: Key Concepts
| Concept | Description |
|---|---|
| Angle Bisector | A line segment or ray that divides an angle into two equal angles. |
| Incenter | The point of concurrency of the three angle bisectors of a triangle. It is the center of the inscribed circle. |
| Formula for Angle at Incenter | \(\angle \text{BOC} = 90^\circ + \frac{\angle \text{A}}{2}\), where O is the incenter and A, B, C are vertices. |
Additional Information: Properties of Incenter
The incenter is a crucial point within a triangle with several interesting properties:
- It is the center of the incircle, which is the largest circle that can be inscribed inside the triangle, tangent to all three sides.
- Every point on an angle bisector is equidistant from the two sides that form the angle. The incenter is the point equidistant from all three sides of the triangle.
- The incenter always lies inside the triangle.
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