Perpendicular Bisector and Angle Bisector Theorem

A perpendicular bisector is a line that divides a line segment into two equal parts at a 90-degree angle.

The Perpendicular Bisector Theorem states that any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment.

An angle bisector is a line or ray that divides an angle into two equal angles.

The Angle Bisector Theorem states that the ratio of the lengths of the two segments created by an angle bisector on the opposite side is equal to the ratio of the lengths of the other two sides of the triangle.

The ratio $$\frac{BD}{DC} = \frac{AB}{AC} = \frac{10}{15} = \frac{2}{3}$$ .

To find the length of a segment using the Perpendicular Bisector Theorem, measure the distance from any point on the bisector to each endpoint of the segment; these distances will be equal.

The angles formed by an angle bisector are equal; if $$\theta$$ is the angle, then both resulting angles are $$\frac{\theta}{2}$$ .

Both angles $$\triangle ABD$$ and $$\triangle ACD$$ measure $$40^\text{o}$$ .

The intersection point is the midpoint of the segment, and it is equidistant from both endpoints.

Using the Angle Bisector Theorem, $$BD = \frac{8}{8+6} \times DC$$ .