Question

Triangle X Y Z is shown. The length of side X Y is (9 n + 12) feet and the length of side X Z is (15 n minus 6) feet. Angles X Y Z and Y Z X are congruent.
Consider isosceles ΔXYZ.

What is the value of n?

What is the measure of leg XY?
ft

What is the measure of leg XZ?
ft

Answer 1

In the given isosceles triangle \( XYZ \), with angles \( \angle XYZ \) and \( \angle YZX \) being congruent, we know that the lengths of sides opposite these angles are equal. Therefore, we can set the lengths of sides \( XY \) and \( XZ \) equal to each other:

\[
XY = XZ
\]

Given the lengths:
- \( XY = 9n + 12 \)
- \( XZ = 15n - 6 \)

Setting these equal gives us the equation:

\[
9n + 12 = 15n - 6
\]

Now, we can solve for \( n \):

\[
12 + 6 = 15n - 9n
\]
\[
18 = 6n
\]
\[
n = 3
\]

Now that we have the value of \( n \), we can find the lengths of sides \( XY \) and \( XZ \):

1. Calculating \( XY \):
\[
XY = 9n + 12 = 9(3) + 12 = 27 + 12 = 39 \text{ feet}
\]

2. Calculating \( XZ \):
\[
XZ = 15n - 6 = 15(3) - 6 = 45 - 6 = 39 \text{ feet}
\]

Since both sides are equal, \( XYZ \) is indeed isosceles.

Finally, the values are:
- The value of \( n \) is \( 3 \).
- The measure of leg \( XY \) is \( 39 \) feet.
- The measure of leg \( XZ \) is \( 39 \) feet.

So, summarizing:
- Value of \( n \): **3**
- Measure of leg \( XY \): **39 ft**
- Measure of leg \( XZ \): **39 ft**