Question

triangle xyz is shown, where n ≥ 5. which statements are true regarding the sides and angles of the triangle? select three options.
□ xy is the longest side.
□ angle x is the largest angle.
□ angle z is greater than angle y.
□ xz is opposite the largest angle.
□ xz is the shortest side.

Explanation:

Step1: Compare side - lengths

Given side - lengths $XY=n + 4$, $YZ = 2n$, and $XZ=n - 2$ with $n\geq5$.
Let's find the differences between the side - lengths.
$YZ - XY=2n-(n + 4)=n - 4$. Since $n\geq5$, $n - 4>0$, so $YZ>XY$.
$XY - XZ=(n + 4)-(n - 2)=6>0$, so $XY>XZ$.
So, $YZ>XY>XZ$. $XZ$ is the shortest side and $YZ$ is the longest side.

Step2: Recall the angle - side relationship in a triangle

In a triangle, the largest angle is opposite the longest side and the smallest angle is opposite the shortest side.
Since $YZ$ is the longest side, the angle opposite it, which is $\angle X$, is the largest angle.
Also, since $XZ\angle Y$.

Answer:

• Angle X is the largest angle.
• Angle Z is greater than angle Y.
• $\overline{XZ}$ is the shortest side.