Are 4 given points a rectangle

CB =

Since DA = CB, it still might be a rectangle. Let's continue by comparing AB and DC. This time I'll use the distance formula just so you can see how it's done.

DA =

CF2 + FB2 = CB2

32 + 12 = CB2

10 = CB2

I made two right triangles by placing points E and F, and connecting them to the vertices of the rectangle. Using the Pythagorean Theorem, we'll check to see if DA is the same length as CB

EA2 + DE2 = DA2

12 + 32 = DA2

10 = DA2

Sure looks like a rectangle, but it might not be.

Let's start by checking the sides. Opposite sides need to be congruent. So lets check and see if DA and CB are congruent. To do this, we can either use the distance formula, or the Pythagorean Theorem. They're actually the same thing, but:

  • Pythagorean Theorem is easier if you can imagine the triangles, or draw on the picture

  • Distance formula is easier if you don't have a picture at all

I'm gonna use Pythagorean Theorem because I like it

  • Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle

For these questions, we'll need to think about the two main properties of rectangles

  • all vertices are right angles

  • opposite sides are congruent

Example 1

Prove whether or not the following 4 points make a rectangle

  • A (2, 1)

  • B (10, 3)

  • C (9, 6)

  • D (1, 4)

We could do this without a picture, but it's always good to make a picture if you don't know 100% what you're doing.

D =

We'll let A (2, 1) be point 1 and B (10,3) be point 2, but we could switch these and still get the same answer.

AB = = = =

For DC, we'll let C (9, 6) be point 1 and D (4, 1) be point 2

DC = = = =

DC = AB, so it still might be a rectangle.

The only thing left to do is to see if the vertices are right angles. In other words, are the sides perpendicular. You learned in GPE 5 that two lines are perpendicular if their slopes multiply to give -1. Let's calculate the slopes of DA and DC to find out.

I prefer the first equation, but you have to think about the negative on your own. The second equation is good if you would rather pick a two points and follow the calculations. We'll use the first one here.

For DA, we'll use the triangle we drew to help us.

But we need to make it negative because it's going up and left

Slope of DA = -3

For DC, we'll draw a triangle to help us

This stays positive because it's going up and right

Now the moment of truth. We multiply our two slopes together to see if they equal -1.

slope of DA X slope of DC = -3 x 1/4 = -3/4

And it doesn't equal -1. That means the two lines aren't perpendicular. Which means they don't meet at a right angle. Which means that this can't be a rectangle

Answer: Not a rectangle, m∠ADC ≠ -1

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