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

For \(n \geq 5\):

• \(2n\) vs \(n+4\): \(2n - (n+4) = n - 4 \geq 1 > 0\) → \(2n > n+4\)
• \(n+4\) vs \(n-2\): \( (n+4)-(n-2)=6>0 \) → \(n+4 > n-2\)

Order: \(2n > n+4 > n-2\) (sides: \(YZ > XY > XZ\))

Step2: Match sides to opposite angles

• \(YZ=2n\) opposite \(\angle X\)
• \(XY=n+4\) opposite \(\angle Z\)
• \(XZ=n-2\) opposite \(\angle Y\)

Step3: Evaluate each statement

1. \(XY\) longest? No (\(YZ\) is longest)
2. \(\angle X\) largest? Yes (opposite longest side \(YZ\))
3. \(\angle Z > \angle Y\)? Yes (\(XY > XZ\) → opposite angles \(\angle Z > \angle Y\))
4. \(XZ\) opposite largest angle? No (largest angle \(\angle X\) opposite \(YZ\))
5. \(XZ\) shortest? Yes (\(n-2\) is smallest side)

Answer:

B. Angle X is the largest angle, C. Angle Z is greater than angle Y, E. XZ is the shortest side