Question

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

Explanation:

Step1: Compare the lengths of the sides

Given \(n\geq5\).

• \(XZ=n - 2\), \(XY=n + 4\), \(YZ = 2n\).
• Let's find the differences:
• \(XY-XZ=(n + 4)-(n - 2)=n+4 - n + 2=6>0\), so \(XY>XZ\).
• \(YZ - XY=2n-(n + 4)=n-4\). Since \(n\geq5\), \(n - 4\geq1>0\), so \(YZ>XY\).
• \(YZ-XZ=2n-(n - 2)=n + 2\). Since \(n\geq5\), \(n + 2\geq7>0\), so \(YZ>XZ\).

Step2: Use the triangle - angle - side relationship

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

• Since \(YZ>XY>XZ\), the angles opposite these sides: \(\angle X>\angle Z>\angle Y\) (because \(\angle X\) is opposite \(YZ\), \(\angle Z\) is opposite \(XY\), \(\angle Y\) is opposite \(XZ\))

Answer:

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