Question

Triangle XYZ is shown, where n ≥ 5.

Triangle X Y Z is shown. The length of side X Y is n + 4, the length of side Y Z is 2 n, and the length of side Z X is n minus 2.

Which statements are true regarding the sides and angles of the triangle? Select three options.

Line segment X Y is the longest side.
Angle X is the largest angle.
Angle Z is greater than angle Y.
Line segment X Z is opposite the largest angle.
Line segment X Z is the shortest side.

Answer 1

Answer:

Angle X is the largest angle.

Angle Z is greater than angle Y.

Line segment X Z is the shortest side.

Step-by-step explanation:

For $n\ge 5,$ you are given

• $XY=n+4;$
• $YZ=2n;$
• $ZX=n-2.$

First,

$XY<XZ+YZ\\ \\XZ<XY+YZ\\ \\YZ<XY+XZ$

So,

$n+4<2n+n-2\Rightarrow 2n>6,\ n>3\\ \\2n<n+4+n-2\Rightarrow 0<2\\ \\n-2<n+4+2n\Rightarrow 2n>-6,\ n>-3$

For $n\ge 5,$

$XY=n+4\ge 9\\ \\YZ=2n\ge 10\\ \\XZ=n-2\ge 3$

So, the greatest side is YZ and the smallest side is XZ. The greatest angle is opposite to the greatest side, then the greatest angle is X. The smallest angle is opposite to the smallest side, so the smallest angle is Y.

Check all options:

A. Line segment X Y is the longest side. - False

B. Angle X is the largest angle. - True

C. Angle Z is greater than angle Y. - True

D. Line segment X Z is opposite the largest angle. - False

E. Line segment X Z is the shortest side. - True.

Answer 2

Angle X is the largest angle, Angle Z is greater than angle Y and Line segment X Z is the shortest side.

What is a triangle?

A triangle is a polygon with three sides and three angles. The sum of any two sides of a triangle is always greater than the third side. The angle of a triangle is proportional to the side opposite the angle.

Given that n ≥ 5, let n = 5, hence:

XY = n + 4 = 5 + 4 = 9

YZ = 2n = 2(5) = 10

XZ = n - 2 = 5 - 2 = 3

∠Z is opposite XY, ∠X is opposite YX and ∠Y is opposite XZ. Hence:

YZ > XY > XZ and ∠X > ∠Z > ∠Y

Angle X is the largest angle, Angle Z is greater than angle Y and Line segment X Z is the shortest side.

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