Question

triangle xyz is shown, where n ≥ 5. image of triangle with vertices x, y, z; side xz labeled n−2, side xy labeled n+4, side yz labeled 2n which statements are true regarding the sides and angles of the triangle? select three options. □ $overline{xy}$ is the longest side. □ angle x is the largest angle. □ angle z is greater than angle y. □ $overline{xz}$ is opposite the largest angle. □ $overline{xz}$ is the shortest side.

Explanation:

Step1: Compare side lengths

Given \( n \geq 5 \), let's compare the lengths of the sides:

• \( XZ = n - 2 \)
• \( XY = n + 4 \)
• \( YZ = 2n \)

Compare \( 2n \) with \( n + 4 \): \( 2n-(n + 4)=n - 4 \). Since \( n\geq5 \), \( n - 4\geq1>0 \), so \( 2n>n + 4 \).

Compare \( n + 4 \) with \( n - 2 \): \( (n + 4)-(n - 2)=6>0 \), so \( n + 4>n - 2 \).

Thus, \( 2n>n + 4>n - 2 \), so \( YZ \) (length \( 2n \)) is the longest side, \( XZ \) (length \( n - 2 \)) is the shortest side.

Step2: Relate sides to angles (larger side opposite larger angle)

• Side \( YZ = 2n \) is opposite angle \( X \), so angle \( X \) is the largest angle (since it's opposite the longest side).
• Side \( XY = n + 4 \) is opposite angle \( Z \).
• Side \( XZ = n - 2 \) is opposite angle \( Y \).

Compare angles: Since \( XY>XZ \) ( \( n + 4>n - 2 \) ), angle \( Z \) (opposite \( XY \)) is greater than angle \( Y \) (opposite \( XZ \)).

Now check each statement:

1. \( \overline{XY} \) is the longest side: False (longest is \( \overline{YZ} \) with length \( 2n \)).
2. Angle \( X \) is the largest angle: True (opposite longest side \( YZ \)).
3. Angle \( Z \) is greater than angle \( Y \): True (since \( XY>XZ \), angle opposite \( XY \) (angle \( Z \)) > angle opposite \( XZ \) (angle \( Y \))).
4. \( \overline{XZ} \) is opposite the largest angle: False (largest angle \( X \) is opposite \( YZ \)).
5. \( \overline{XZ} \) is the shortest side: True (as \( n - 2 \) is the smallest length).

Answer:

Angle \( X \) is the largest angle, Angle \( Z \) is greater than angle \( Y \), \( \overline{XZ} \) is the shortest side (i.e., the three true statements are: "Angle \( X \) is the largest angle", "Angle \( Z \) is greater than angle \( Y \)", "\( \overline{XZ} \) is the shortest side").