Question

a triangular garden is to be split so that the angle at vertex b is bisected. this diagram was supplied by the landscape architect, but you do not have a way to measure the angles at b. you do have the given side lengths, so what is the length of side ad that will allow the angle at b to be bisected?

a. 4m
b. 3m
c. 5m
d. 2m

Explanation:

Step1: Apply the Angle - Bisector Theorem

The Angle - Bisector Theorem states that if a bisector of an angle of a triangle divides the opposite side into segments proportional to the adjacent sides. For \(\triangle ABC\) with angle bisector \(BD\), we have \(\frac{AD}{DC}=\frac{AB}{BC}\).
Given \(AB = 6m\), \(BC=8m\), and \(DC = 4m\). Substitute these values into the proportion \(\frac{AD}{4}=\frac{6}{8}\).

Step2: Solve the proportion for \(AD\)

Cross - multiply the proportion \(\frac{AD}{4}=\frac{6}{8}\). We get \(8AD=6\times4\).
Simplify the right - hand side: \(6\times4 = 24\). So, \(8AD = 24\).
Divide both sides by 8: \(AD=\frac{24}{8}=3\).

Answer:

B. \(3m\)