# Unlocking the Secrets of Triangle Angle Bisectors

When it comes to triangles, the intrigue doesn't just lie in their sides and vertices, but also in the invisible lines that give them character. One such line is the 'angle bisector'. An angle bisector in a triangle does precisely what its name implies: it cuts an angle into two equal parts. Dive with us into the world of triangle angle bisectors, where we'll unravel their properties, applications, and the underlying mathematics.

## Step-by-step Guide: Angle Bisectors of Triangles

**1. Definition of Angle Bisector:**

An angle bisector in a triangle is a ray or segment that divides an interior angle of the triangle into two congruent or equal angles. Every triangle has three angle bisectors, one for each interior angle.

**2. Properties of Angle Bisectors:**

- 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.

**3. Formulae:**

Given a triangle \(ABC\) with sides \(a\), \(b\), and \(c\), and if \(AD\) is the angle bisector of angle \(A\), dividing the opposite side \(BC\) into segments of length \(x\) and \(y\), then:

\( \frac{x}{a} = \frac{y}{b} \)

### Examples

**Example 1:**

In triangle \(PQR\), side \(PQ = 6 \text{ cm}\), side \(QR = 8 \text{ cm}\), and side \(RP = 10 \text{ cm}\). If RS is the angle bisector of angle \(Q\), and it divides side \(PR \) into segments of \(4 \text{ cm}\) and \(6 \text{ cm}\), verify the bisector theorem.

**Solution:**

According to the angle bisector theorem:

\( \frac{4 \text{ cm}}{6 \text{ cm}} = \frac{RS}{8 \text{ cm}} \Rightarrow \frac{2}{3} = \frac{RS}{8 \text{ cm}} \)

Thus, \( RS = \frac{2}{3} \times 8 \text{ cm} = 5.33 \text{ cm} \), which verifies the theorem.

**Example 2:**

Given triangle \(XYZ \) with sides \(XZ = 9 \text{ cm}\), side \(XY = 7.5 \text{ cm}\), and angle bisector \(ZY \) divides side \(XX \) into segments \(4 \text{ cm}\) and \(5.5 \text{ cm}\). Find the length of side \(YZ \).

**Solution:**

Using the angle bisector theorem:

\( \frac{4 \text{ cm}}{9 \text{ cm}} = \frac{5.5 \text{ cm}}{YZ} \)

On solving, we get \( YZ = 12.375 \text{ cm} \).

### Practice Questions:

- 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 \(AC\).
- 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 \(EF\).

**Answers:**

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

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