Question

In a triangle ABC, AD is the bisector of angle A. If AC =4.2 cm, DC = 6 cm, BC =10 cm, find AB.

• A. 2.7 cm
• B. 2.8 cm
• C. 3.4 cm
• D. 2.9 cm

Answer 1

The Correct Option is B

To find the length of side AB in triangle ABC where AD is the angle bisector of angle A, we can apply the Angle Bisector Theorem. According to the theorem, when an angle bisector divides a triangle, the lengths of the two segments created on the opposite side are proportional to the other two sides of the triangle.

In triangle ABC, AD is the angle bisector, dividing side BC into two segments: \(\overline{BD}\) and \(\overline{DC}\). According to the Angle Bisector Theorem:

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

Given: 

• \(AC = 4.2 \, \text{cm}\)
• \(DC = 6 \, \text{cm}\)
• \(BC = 10 \, \text{cm}\)

Therefore, \(BD = BC - DC = 10 - 6 = 4 \, \text{cm}\)

Substituting into the equation from the Angle Bisector Theorem:

\(\frac{AB}{4.2} = \frac{4}{6}\)

Simplifying the right side:

\(\frac{AB}{4.2} = \frac{2}{3}\)

Cross-multiplying to solve for \(AB\):

\(AB = \frac{2}{3} \times 4.2\)

Calculating:

\(AB = \frac{8.4}{3} = 2.8 \, \text{cm}\)

Thus, the length of side AB is 2.8 cm, which matches the correct answer option.