Question

applying the converse of the isosceles triangle theorem
consider isosceles \\( \triangle x y z \\).
what is the value of \\( n \\)?
what is the measure of leg \\( x y \\)? \\( \mathrm{ft} \\)
what is the measure of leg \\( x z \\)? \\( \mathrm{ft} \\)

Explanation:

Step1: Use the property of isosceles triangle

In an isosceles triangle, if two angles are equal, then the sides opposite to them are equal. Here, \(XY = XZ\). So, \(9n + 12=15n - 6\).

Step2: Solve the equation for \(n\)

Subtract \(9n\) from both sides: \(12 = 15n-9n - 6\), which simplifies to \(12=6n - 6\).
Add \(6\) to both sides: \(12 + 6=6n\), so \(18 = 6n\).
Divide both sides by \(6\): \(n=\frac{18}{6}=3\).

Step3: Find the length of \(XY\)

Substitute \(n = 3\) into the expression for \(XY\): \(XY=9n + 12\).
\(XY=9\times3+12=27 + 12=39\) (ft).

Step4: Find the length of \(XZ\)

Substitute \(n = 3\) into the expression for \(XZ\): \(XZ=15n - 6\).
\(XZ=15\times3-6=45 - 6=39\) (ft).

Answer:

The value of \(n\) is \(3\).
The measure of leg \(XY\) is \(39\) ft.
The measure of leg \(XZ\) is \(39\) ft.