Question

Which statement proves that △XYZ is an isosceles right triangle?

Answer 1

In order for the triangle to be called an isosceles right triangle, it must satisfy both conditions: XZ and XY must be perpendicular to each other, and XZ = XY.
To determine whether two lines are perpendicular, their slopes must be negative reciprocals of each other.

Solving for the slopes of XZ and XY:
slope of XZ = (y₂-y₁) / (x₂-x₁) = (6-3)/(5-1) = 3/4
slope of XY = (y₂-y₁) / (x₂-x₁) = (3--1)/(1-4) = -4/3
-4/3 is the negative reciprocal of 3/4, therefore XZ and XY are perpendicular to each other.

Solving for distance:
XZ = √[(y₂-y₁)²+(x₂-x₁)²] = √[(6-3)²+(5-1)²] = 5
XY = √[(y₂-y₁)²+(x₂-x₁)²] = √[(3--1)²+(1-4)²] = 5

The slope of XZ is 3/4, the slope of XY is -4/3, and XZ = XY = 5. 

Answer 2

The correct answer is:

C. The slope of XZ is $\frac{3}{4}$, the slope of XY is $-\frac{4}{3}$, and XZ = XY = 5.

The statement above proves that △XYZ is an isosceles right triangle.

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