Question

Determine whether quadrilateral ABCD is a rhombus, a rectangle, a square, a parallelogram, or none. List all that apply. Explain.

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Help pls? anyone :(?

Answer 1

• ABCD is a rhombus, and a parallelogram

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Given

• Points A(-6, - 1), B(4, - 6), C(2, 5), D(- 8, 10)

First, plot the points (see attached picture).

Then, connect all the points.

We see that:

• Opposite sides are parallel,
• Diagonals are perpendicular.

From our observation the figure is rhombus.

Let's confirm it with the following.

1) Find midpoints of diagonals and compare.

• AC → x = (- 6 + 2)/2 = - 2, y = (- 1 + 5)/2 = 2
• BD → x = (4 - 8)/2 = - 2, y = (- 6 + 10)/2 = 2

The midpoint of both diagonals is same (- 2, 2).

2) Find slopes of diagonals and check if their product is -1, this will confirm they are perpendicular.

• m(AC) = (5 - (-1))/(2 - (-6)) = 6/8 = 3/4
• m(BD) = (10 - (-6))/(-8 - 4) = - 16/12 = - 4/3

• m(AC) × m(BD) = 3/4 * (- 4/3) = - 1

Confirmed.

So this is a rhombus and also a parallelogram but not rectangle or square, since opposite angles are not right angles.