Sorry, another question.
** Determine whether parallelogram ABCD is a rhombus, a rectangle, or a square. List all that apply. Explain your reasoning.
1) A(12,0), B(6,-6), C(0,0), D(6,6)
I know to do this. I just don't know if it's a rhombus, rectangle, or a square. For AB I got -6/-5; BC= 6/-6; CD= 6/6; and AD= 6/5..........is this all right? And how can I tell if it's either one of these (any easy way to tell)?
To determine whether parallelogram ABCD is a rhombus, a rectangle, or a square, there are a few steps you can follow:
Step 1: Calculate the slopes of the opposite sides.
- The slope of side AB can be calculated as (y2 - y1) / (x2 - x1) using the coordinates of points A(12,0) and B(6,-6).
- The slope of side BC can be calculated using the coordinates of points B(6,-6) and C(0,0).
- The slope of side CD can be calculated using the coordinates of points C(0,0) and D(6,6).
- The slope of side AD can be calculated using the coordinates of points A(12,0) and D(6,6).
Step 2: Determine if the opposite sides are parallel.
- If the slopes of sides AB and CD are equal, and the slopes of sides BC and AD are equal, then the opposite sides are parallel.
Step 3: Determine if the opposite sides are equal in length.
- If the lengths of sides AB and CD are equal, and the lengths of sides BC and AD are equal, then the opposite sides are equal.
Based on the calculations you provided, let's check if it's a rhombus, a rectangle, or a square:
- AB: (-6 - 0) / (6 - 12) = -6 / -6 = 1
- BC: (0 - (-6)) / (0 - 6) = 6 / -6 = -1
- CD: (6 - 0) / (6 - 0) = 6 / 6 = 1
- AD: (6 - 0) / (6 - 12) = 6 / -6 = -1
From the slopes, we can see that AB is parallel to CD and BC is parallel to AD, satisfying Step 2.
However, for Step 3, we need to also check if the opposite sides are equal in length. Calculating the length of each side using the distance formula:
- AB: √[(6 - 12)^2 + (-6 - 0)^2] = √36 + 36 = √72 ≈ 8.49
- BC: √[(0 - 6)^2 + (0 - (-6))^2] = √36 + 36 = √72 ≈ 8.49
- CD: √[(6 - 0)^2 + (6 - 0)^2] = √36 + 36 = √72 ≈ 8.49
- AD: √[(6 - 12)^2 + (6 - 0)^2] = √36 + 36 = √72 ≈ 8.49
Since all the opposite sides have the same length, ABCD satisfies Step 3 as well.
From the calculations, we can conclude that parallelogram ABCD is both a rhombus and a rectangle. It is a rhombus because all sides have equal lengths, and it is a rectangle because all angles are right angles. However, it is not a square because it does not have all sides and angles equal.