What is the solution to this system of linear equations: y-x=4 and y+2x=1
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Answer:
- ABCD is a rhombus, and a parallelogram
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<h3>Given </h3>- 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.
<h3>We see that:</h3>- 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
<u>Confirmed.</u>
So this is a rhombus and also a parallelogram but <u>not</u> rectangle or square, since opposite angles are not right angles.
