If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.
Here's a detailed explanation:
1. Understanding the Angle Bisector:
- An angle bisector is a line or ray that divides an angle into two congruent angles. In other words, each of the two angles formed is equal in measure.
2. Identifying the Point:
- Suppose we have an angle [tex]\( \angle ABC \)[/tex] and a point [tex]\( P \)[/tex] lies on the angle bisector of [tex]\( \angle ABC \)[/tex].
3. Proving Equidistance:
- Since [tex]\( P \)[/tex] is on the angle bisector, it means that [tex]\( PA \)[/tex] and [tex]\( PB \)[/tex] are the perpendicular distances from the point [tex]\( P \)[/tex] to the sides [tex]\( AB \)[/tex] and [tex]\( BC \)[/tex] of [tex]\( \angle ABC \)[/tex].
- By the property of angle bisectors, the distance from any point on the bisector of an angle to the sides of the angle are equal.
4. Conclusion:
- Therefore, if a point [tex]\( P \)[/tex] is on the bisector of [tex]\( \angle ABC \)[/tex], then the distances from [tex]\( P \)[/tex] to the sides [tex]\( AB \)[/tex] and [tex]\( BC \)[/tex] must be the same.
- This is a fundamental property of angle bisectors and is often used in various geometrical proofs and constructions.
So, the key takeaway is:
If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.
