Question

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

Explanation:

Step1: Analyze side lengths

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

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

Compare \( n - 2 \), \( n + 4 \), and \( 2n \):
For \( n \geq 5 \), \( 2n-(n + 4)=n - 4 \geq 1 \) (since \( n \geq 5 \)), so \( 2n > n + 4 \).
\( n + 4-(n - 2)=6 > 0 \), so \( n + 4 > n - 2 \). Thus, \( 2n > n + 4 > n - 2 \), so \( YZ \) is the longest side, \( XZ \) is the shortest side.

Step2: Analyze angles (larger side opposite larger angle)

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

Step3: Evaluate each statement

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

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").