Question

select the correct answer.
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. 5m
b. 4m
c. 2m
d. 3m

Explanation:

Step1: Apply 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. So, $\frac{AD}{DC}=\frac{AB}{BC}$.
Given $AB = 6\,\text{m}$, $BC = 8\,\text{m}$, $DC = 4\,\text{m}$, and $AD = x\,\text{m}$. Substitute into the theorem: $\frac{x}{4}=\frac{6}{8}$.

Step2: Solve for \( x \)

Cross - multiply: $8x=6\times4$.
Simplify the right - hand side: $8x = 24$.
Divide both sides by 8: $x=\frac{24}{8}=3$. Wait, no, wait. Wait, let's re - check. Wait, the Angle Bisector Theorem is $\frac{AD}{DC}=\frac{AB}{BC}$. Wait, $AB = 6$, $BC = 8$, $DC = 4$, $AD=x$. So $\frac{x}{4}=\frac{6}{8}$. Cross - multiply: $8x=24$, $x = 3$? But wait, maybe I mixed up the sides. Wait, no, the Angle Bisector Theorem: In $\triangle ABC$, if $BD$ bisects $\angle ABC$, then $\frac{AD}{DC}=\frac{AB}{BC}$. So $AB = 6$, $BC = 8$, $DC = 4$, so $\frac{x}{4}=\frac{6}{8}$, $x=\frac{6\times4}{8}=\frac{24}{8}=3$? But the options have 3 as option D. Wait, but let's check again. Wait, maybe I made a mistake. Wait, $AB = 6$, $BC = 8$, $AD=x$, $DC = 4$. So $\frac{x}{4}=\frac{6}{8}$, $x = 3$. So the length of $AD$ is 3m.

Answer:

D. 3m