Triangle Geometry: Angle Bisector Theorem Application

This problem requires applying the Angle Bisector Theorem to a triangle ABC where the angle bisector of \(\angle \text{BAC}\) divides the opposite side BC.

Angle Bisector Theorem Explanation

The Angle Bisector Theorem states that if a line bisects an angle of a triangle, it divides the opposite side into two segments that are proportional to the other two sides of the triangle. For triangle ABC, with AD bisecting \(\angle \text{BAC}\):

\(\frac{AB}{AC} = \frac{BD}{DC}\)

Applying the Theorem to Triangle ABC

Given:

• AB = 18 cm
• AC = 15 cm
• BC = 22 cm

Using the Angle Bisector Theorem:

\(\frac{18}{15} = \frac{BD}{DC}\)

Simplify the ratio:

\(\frac{6}{5} = \frac{BD}{DC}\)

This means the ratio \(BD:DC\) is \(6:5\). Since \(BD + DC = BC = 22\) cm, we can find the lengths of BD and DC.

Calculating Segment Lengths

The total ratio parts are \(6 + 5 = 11\).

Calculate BD:

\(BD = \left(\frac{6}{11}\right) \times BC = \left(\frac{6}{11}\right) \times 22 \text{ cm} = 6 \times 2 \text{ cm} = 12 \text{ cm}\)

Calculate DC:

\(DC = \left(\frac{5}{11}\right) \times BC = \left(\frac{5}{11}\right) \times 22 \text{ cm} = 5 \times 2 \text{ cm} = 10 \text{ cm}\)

We can check that \(BD + DC = 12 + 10 = 22\) cm, which is correct.

Final Calculation: Product of Segments

The question asks for the value of (BD × DC).

\(BD \times DC = 12 \text{ cm} \times 10 \text{ cm} = 120 \text{ cm}^2\)