Unlocking the Secrets of Triangle Angle Bisectors

An angle bisector cuts an angle into two equal halves. In a triangle, the bisector of an angle divides the opposite side into two segments proportional to the two adjacent sides.

Free tool: try the problems from this lesson in our Arc Length & Sector Area Calculator.

• The angle bisector divides the opposite side into two segments that are proportional to the other two sides of the triangle.
• The three angle bisectors of a triangle are concurrent; they meet at a single point called the “incenter” of the triangle. This point is equidistant from all three sides and is the center of the inscribed circle (or incircle) of the triangle.

Worked examples: triangle angle bisectors

Practice questions on angle bisectors in triangles

1. In triangle \(ABC \), \(BA = 5 \text{ cm}\), \(BC = 7 \text{ cm}\), and the angle bisector \(AD \) divides side \(BC \) into lengths of \(3 \text{ cm}\) and \(4 \text{ cm}\). Find the length of side \(AC\).
2. For triangle \(DEF\), if \(DE = 8 \text{ cm}\), angle bisector \(DF\) divides side \(DE\) into lengths of \(5 \text{ cm}\) and \(3 \text{ cm}\), determine the length of side \(EF\).

1. Using the angle bisector theorem, \( AC = \frac{5 \times 4}{3} = 6.67 \text{ cm}\).
2. \( EF = \frac{8 \times 3}{5} = 4.8 \text{ cm}\).

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