Short Answer 
A right isosceles triangle has one angle of 90 degrees and two equal sides. To confirm triangle XYZ is such, one must calculate the slopes of its sides, XZ and XY, which yield perpendicular lines, and verify that the lengths of both sides are equal.
Step 1: Understand the Definition of Right Isosceles Triangle
A right isosceles triangle is characterized by having one angle that is 90 degrees and two sides that are equal in length. In this case, the two equal sides correspond to the right angle, creating a balanced triangle structure. This means that triangle XYZ has sides XZ and XY which are equal.
Step 2: Calculate the Slopes of the Lines
To determine if triangle XYZ is a right isosceles triangle, we must calculate the slopes of the lines XZ and XY. We use the slope formula: m = (y‚ÄöCC – y‚ÄöCA)/(x‚ÄöCC – x‚ÄöCA). For lines XZ and XY, the calculations are as follows:
- Slope of XZ = (6 – 3) / (5 – 1) = 3/4
- Slope of XY = (3 – (-1)) / (1 – 4) = -4/3
Step 3: Confirm Perpendicularity and Equality of Sides
Once the slopes are calculated, check if the product of the slopes equals -1 to confirm that the lines are perpendicular. Since 3/4 * -4/3 = -1, XZ and XY are indeed perpendicular. Furthermore, both lengths are found to be 5, confirming that triangle XYZ is a right isosceles triangle.