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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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