Question

find the value of the variable.
(image of a triangle with sides 3, 4.5, 1.5, and a segment creating a smaller triangle with side z; z = blank)

Explanation:

Step1: Apply the Angle - Bisector Theorem

The Angle - Bisector Theorem states that if a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the adjacent sides. So, we have the proportion \(\frac{3}{z}=\frac{4.5}{1.5}\).

Step2: Cross - multiply

Cross - multiplying gives \(4.5z = 3\times1.5\).

Step3: Simplify the right - hand side

\(3\times1.5 = 4.5\), so the equation becomes \(4.5z=4.5\).

Step4: Solve for \(z\)

Divide both sides of the equation \(4.5z = 4.5\) by \(4.5\). We get \(z=\frac{4.5}{4.5}\).

Answer:

\(1\)