The coordinates of this rectangle WXYZ is W(-23) X(1,6) Y(3,4) Z(0,1)

Question

Use slopes to prove this is a rectangle. What is the slope of side XY? What is the slope of WX? Are these sides parallel or perpendicular ?
A. Slope of XY=-2, slope of WX = 2 these lines are perpendicular
B. slope of XY =-2 slope of WX = 2, these lines are parallel
C. Slope of XY=-1, slope of WX=1, these lines are parallel
D. Slope of XY =-1, slope of WX =1, these lines are perpendicular

Answer 1

To determine whether rectangle WXYZ is indeed a rectangle, we can calculate the slopes of sides XY and WX and then check if these sides are perpendicular (which is characteristic of rectangles).

Coordinates:

• W = (-2, 3)
• X = (1, 6)
• Y = (3, 4)
• Z = (0, 1)

Step 1: Calculate the slopes.

Slope formula: \[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]

Slope of side XY: Coordinates of X (1, 6) and Y (3, 4): \[ \text{slope of XY} = \frac{4 - 6}{3 - 1} = \frac{-2}{2} = -1 \]

Slope of side WX: Coordinates of W (-2, 3) and X (1, 6): \[ \text{slope of WX} = \frac{6 - 3}{1 - (-2)} = \frac{3}{3} = 1 \]

Step 2: Determine if the sides are parallel or perpendicular.

• Two lines are parallel if their slopes are equal.
• Two lines are perpendicular if the product of their slopes equals -1.

Now, let's analyze the slopes we found:

• Slope of XY = -1
• Slope of WX = 1

Checking for perpendicularity: \[ \text{slope of XY} \cdot \text{slope of WX} = -1 \cdot 1 = -1 \] Since their product is -1, WX and XY are perpendicular.

Conclusion

The slopes we calculated are:

• Slope of XY = -1
• Slope of WX = 1 Therefore, the correct answer is D. Slope of XY = -1, slope of WX = 1, these lines are perpendicular.