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.
In \(\triangle ABC\) with \(BD\) bisecting \(\angle ABC\), we have \(\frac{AD}{DC}=\frac{AB}{BC}\).
Let \(AD = x\), \(AB = 6\space m\), \(BC = 8\space m\), and \(DC=4\space m\).
Substituting the values into the formula \(\frac{x}{4}=\frac{6}{8}\).

Step2: Solve the proportion for \(x\)

Cross - multiply the proportion \(\frac{x}{4}=\frac{6}{8}\).
We get \(8x=4\times6\).
Since \(4\times6 = 24\), the equation becomes \(8x = 24\).
Divide both sides by \(8\): \(x=\frac{24}{8}=3\).

Answer:

B. \(3m\)