Question

triangle xyz is shown, where ( ngeq5 ).
which statements are true regarding the sides and angles of the triangle? select three options.
( square overline{xz} ) is the shortest side.
( square overline{xz} ) is opposite the largest angle.
( square ) angle ( z ) is greater than angle ( y ).
( square ) angle ( x ) is the largest angle.
( square overline{xy} ) is the longest side.

Explanation:

Step1: Analyze side lengths

Given \( n \geq 5 \), let's compare the side lengths:
- \( XZ = n - 2 \)
- \( XY = n + 4 \)
- \( YZ = 2n \)

For \( n \geq 5 \), let's take \( n = 5 \) as a test case:
- \( XZ = 5 - 2 = 3 \)
- \( XY = 5 + 4 = 9 \)
- \( YZ = 2\times5 = 10 \)

In general, compare \( 2n \), \( n + 4 \), and \( n - 2 \):
- \( 2n - (n + 4) = n - 4 \). Since \( n \geq 5 \), \( n - 4 \geq 1 > 0 \), so \( 2n > n + 4 \).
- \( n + 4 - (n - 2) = 6 > 0 \), so \( n + 4 > n - 2 \). Thus, \( YZ = 2n \) is the longest side? Wait, no, wait the options: Wait, the options have " \( \overline{XY} \) is the longest side" – no, wait our calculation shows \( YZ = 2n \) is longer. Wait, maybe I misread the sides. Wait the triangle: \( XZ \) is \( n - 2 \), \( XY \) is \( n + 4 \), \( YZ \) is \( 2n \). Wait, let's re - check the side opposite angles.

Angle opposite a side: In triangle \( XYZ \), angle \( X \) is opposite \( YZ \) (length \( 2n \)), angle \( Y \) is opposite \( XZ \) (length \( n - 2 \)), angle \( Z \) is opposite \( XY \) (length \( n + 4 \)).

Now, let's analyze each option:

1. " \( \overline{XY} \) is the longest side": \( XY = n + 4 \), \( YZ = 2n \). For \( n \geq 5 \), \( 2n - (n + 4)=n - 4\geq1 \), so \( YZ > XY \). So this is false.

2. "Angle \( X \) is the largest angle": Angle \( X \) is opposite \( YZ = 2n \), which is the longest side (since \( 2n > n + 4 > n - 2 \) for \( n \geq 5 \)). In a triangle, the largest angle is opposite the longest side. So angle \( X \) is opposite the longest side \( YZ \), so angle \( X \) is the largest angle. This is true.

3. "Angle \( Z \) is greater than angle \( Y \)": Angle \( Z \) is opposite \( XY = n + 4 \), angle \( Y \) is opposite \( XZ = n - 2 \). Since \( n + 4 > n - 2 \), by the "larger side opposite larger angle" theorem, angle \( Z \) (opposite \( n + 4 \)) is greater than angle \( Y \) (opposite \( n - 2 \)). This is true.

4. " \( \overline{XZ} \) is opposite the largest angle": The largest angle is opposite the longest side \( YZ = 2n \), which is opposite angle \( X \), not angle opposite \( XZ \). So this is false.

5. " \( \overline{XZ} \) is the shortest side": \( XZ = n - 2 \), \( XY = n + 4 \), \( YZ = 2n \). As \( n - 2 < n + 4 < 2n \) for \( n \geq 5 \), \( XZ \) is the shortest side. This is true. Wait, but the problem says "Select three options". Wait maybe my initial test case was wrong. Wait let's re - evaluate:

Wait the options:

- \( \overline{XY} \) is the longest side: False (as \( YZ = 2n \) is longer)
- Angle \( X \) is the largest angle: True (opposite longest side \( YZ \))
- Angle \( Z \) is greater than angle \( Y \): True (opposite \( XY = n + 4 \) which is longer than \( XZ = n - 2 \))
- \( \overline{XZ} \) is opposite the largest angle: False (largest angle opposite \( YZ \))
- \( \overline{XZ} \) is the shortest side: True (since \( n - 2 < n + 4 < 2n \) for \( n \geq 5 \))

Wait, but the problem says "Select three options". Wait maybe I made a mistake in side labeling. Let's re - look at the triangle:

Points \( X \), \( Y \), \( Z \). \( XZ \) is \( n - 2 \), \( XY \) is \( n + 4 \), \( YZ \) is \( 2n \).

Wait, angle opposite \( XZ \) (side \( XZ \)) is angle \( Y \), angle opposite \( XY \) (side \( XY \)) is angle \( Z \), angle opposite \( YZ \) (side \( YZ \)) is angle \( X \).

Now, let's check the length of sides:

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

For \( n = 5 \):

\( XZ = 3 \), \( XY = 9 \), \( YZ = 10 \)

So side lengths: \( XZ = 3 \), \( XY = 9 \), \( YZ = 10 \). So order: \( XZ < XY < YZ \)

So:

- \( \overline{XZ} \) is the shortest side: True (3 is the shortest)
- Angle \( X \) is opposite \( YZ = 10 \) (longest side), so angle \( X \) is the largest angle: True
- Angle \( Z \) is opposite \( XY = 9 \), angle \( Y \) is opposite \( XZ = 3 \). Since \( 9 > 3 \), angle \( Z \) (opposite 9) is greater than angle \( Y \) (opposite 3): True
- \( \overline{XY} \) is the longest side: False (YZ is 10 > 9)
- \( \overline{XZ} \) is opposite the largest angle: False (largest angle opposite YZ)

So the three true statements are:

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

But let's confirm with the triangle side - angle relationship (larger side opposite larger angle):

- Side \( YZ = 2n \) (longest) opposite angle \( X \) → angle \( X \) is largest: True
- Side \( XY = n + 4 \) (middle) opposite angle \( Z \), side \( XZ = n - 2 \) (shortest) opposite angle \( Y \). Since \( XY > XZ \), angle \( Z > \) angle \( Y \): True
- Side \( XZ = n - 2 \) is the shortest (since \( n - 2 < n + 4 < 2n \) for \( n \geq 5 \)): True

Answer:

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