Question

triangle xyz is shown, where n ≥ 5. which statements are true regarding the sides and angles of the triangle? choose three correct answers. angle z is greater than angle y. angle x is the largest angle. \\(\overline{xz}\\) is the shortest side. \\(\overline{xz}\\) is opposite the largest angle. \\(\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 \( n = 5 \) (minimum value) to test:

• \( XZ = 5 - 2 = 3 \)
• \( XY = 5 + 4 = 9 \)
• \( YZ = 2 \times 5 = 10 \)

So, \( XZ < XY < YZ \) (since \( 3 < 9 < 10 \)). In general, for \( n \geq 5 \):

• \( n - 2 < n + 4 \) (because \( -2 < 4 \))
• \( n + 4 < 2n \) (since \( 4 < n \) as \( n \geq 5 \))

Thus, the order of sides is \( XZ < XY < YZ \).

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

In a triangle, the larger side is opposite the larger angle.

• Side \( XZ \) (length \( n - 2 \)) is opposite angle \( Y \).
• Side \( XY \) (length \( n + 4 \)) is opposite angle \( Z \).
• Side \( YZ \) (length \( 2n \)) is opposite angle \( X \).

From side order \( XZ < XY < YZ \), the angle order (opposite) is \( \angle Y < \angle Z < \angle X \).

Step3: Evaluate each statement

1. "Angle Z is greater than angle Y": Since \( \angle Z > \angle Y \) (from angle order), this is true.
2. "Angle X is the largest angle": Since \( \angle X \) is opposite the longest side \( YZ \), this is true.
3. " \( \overline{XZ} \) is the shortest side": From side order \( XZ < XY < YZ \), this is true.
4. " \( \overline{XZ} \) is opposite the largest angle": \( XZ \) is opposite \( \angle Y \), which is the smallest angle, so false.
5. " \( \overline{XY} \) is the longest side": \( YZ \) is the longest side, so false.

Answer:

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