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

We have side lengths \(XY=n + 4\), \(XZ=n - 2\), and \(YZ = 2n\). Given \(n\geq5\).
Let's find the differences between the side - lengths.
First, compare \(YZ\) and \(XY\): \(YZ - XY=2n-(n + 4)=n - 4\). Since \(n\geq5\), \(n - 4>0\), so \(YZ>XY\).
Next, compare \(XY\) and \(XZ\): \(XY - XZ=(n + 4)-(n - 2)=6>0\), so \(XY>XZ\). So the order of side - lengths from shortest to longest is \(XZ

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.
The longest side is \(YZ\), and the angle opposite \(YZ\) is \(\angle X\). So \(\angle X\) is the largest angle.
The shortest side is \(XZ\).
The side opposite \(\angle Z\) is \(XY\) and the side opposite \(\angle Y\) is \(XZ\). Since \(XY>XZ\), by the angle - side relationship (\(a>b\) implies \(\angle A>\angle B\) where \(a\) and \(b\) are side - lengths and \(\angle A\) and \(\angle B\) are the opposite angles), \(\angle Z>\angle Y\).

Answer:

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