Quadrilaterals - Rhombus

Question

Verify that the shape with vertices at P(-5,2), Q(-1,3), R(-2,-1) and S(-6,-2) is a rhombus

Answer 1

By getting the differences between the x's and y's; and they're =; the lines are parallel.

P-S=1,4

Q-R=1,4

Answer 2

Probably the easiest way to do this is to graph each point first. If you graph each point, you will see that that

S = Bottom Left point

P = Upper left point

R = Bottom Right point

Q = Upper right point

Now we need to know the definition of a rhombus. A rhombus is any parallelogram with equal sides. The sides in this case are

PS

PQ

QR

SR

The easiest method would be to count the distance of each point from each other, for each side, or to use the distance formula. For this answer, we will use the counting method. When we count, the points for each side should be the same distance.

If you count each side, you will see that for SR and PQ, they are both 8 units away from each other horizontally and 2 units away from each other vertically

If you count PS and QR, you will find that they are 8 units away from each other vertically and 2 units away from each other horizontally.

Without doing anything further, we know that this is a rhombus because the horizontal sides and vertical sides ( and each side individually) are equal to each other.

Answer 3

The length of all four sides must be equal. Start by plotting the coordinates and then finding the length of each side using the distance formula d=sqrt[(X2-X1)^2+(Y2-Y1)^2 (unless the shape is obviously not a rhombus, if it's not symmetric, etc)

Answer 4

Plot the points and you should identify this as a rhombus by joining the points with lines and see that it has two sets of parallel sides.

https://www.google.com/search?q=rhombus&rlz=1C1GCEA_enUS863US863&oq=rhombus&aqs=chrome..69i57j0l5.4435j1j8&sourceid=chrome&ie=UTF-8

"A Rhombus is a flat shape with 4 equal straight sides. A rhombus looks like a diamond. All sides have equal length. Opposite sides are parallel, and opposite angles are equal (it is a Parallelogram)."

To verify that the two sets of parallel sides are indeed parallel, you should verify that the slopes of these sides are equal.

To do this, take the two points that join each side and find the slope m = (y2 - y1)/(x2 - x1). When you do this, if there are two sets of matching slopes, then you indeed have a rhombus.

You should also verify that each side has equal length. To do this, use the distance formula given here:

https://www.wikihow.com/Find-the-Distance-Between-Two-Points

All that is required for this is to identify one vertex of each side as (x1,y1) and the other at (x2,y2) and plug into the formula. Do this four times, once for each of the four combinations of vertices.

Please send me a message if you have any questions or want me to work out the proof in full detail.

Good luck :)

John B.