Question

Help needed please! The quadrilateral ABCD, where A is (4,5) and C is (3,-2), is a square. Find the coordinates of B and D and hence or otherwise the area of the square.

Answer 1

Hmm, I think you can find the area of the square w/o knowing the other coordinates, just the length of diagonal is enough. But it's good to know both methods.

Do you know how to find the distance between two points?

Answer 2

Yes, I do.

Answer 3

I've found the length of the diagonal and thus the area but I don't know how to find two missing coordinates.

Answer 4

Ah, okay well... fun. Finding the other two points is not actually very easy, since the two points we have aren't nicely lined up, they look like in the image.

You might have to use... okay, maybe use the fact that two lines that pass through A and C are perpendicular and thus their slopes are related by \[m _{2} = \frac{ -1 }{ m _{1} }\] and find the point where they intersect.

Answer 5

So I have to find the point of intersection between A and C to find the coordinates?

Answer 6

Okay so I have one of the equation which is x + 7y = 39 which I got from AC.

Answer 7

But where would the other equation come from? I need to solve it as a simultaneous equation so that I can find points of intersection.

Answer 8

Well we have two lines, lets call 'em yA and yB, yA passes through point A, yB passes through point B.

\[y _{A} = m _{A} x + b _{A}\]
\[y _{B} = m _{B} x + b _{B}\]

Their gradients are related by \[m _{A} = \frac{ -1 }{m _{B} }\]

Answer 9

I'm not 100% sure this alone will let us solve it, we may also need to use the distance formula, but i'm hoping not.

Answer 10

yA passes through 4,5
\[5 = 4 m _{A} + b _{A} \]
yB through (3, -2)
\[3 = -2\frac{-1}{m _{A} } + b _{B} \]
Dammit, we have three variables and two equations so we may need the dist. formula too.

Answer 11

What do we have to find the distance for?

Answer 12

Ohhh yeah I see why it didn't work... forgot that there's two lines that pass through A and C, that are perpendicular... the ones that pass through B and the ones that pass through D.

Answer 13

The distances AB, CB, CD, AD are all equal. And with the perpendicular lines, there's enough to solve it.

Answer 14

Thank you for actually helping me step by step by the way.

Answer 15

No prob. I was just hoping to find an easier method this time, solved a square similar to this yesterday.

Answer 16

So will I have to use the distance formula? I'm still confused as to what I should do to find B and D.

Answer 17

Let's try this, using the gradients
slope from A to B is:\[\frac{ (y-5)}{(x-4)}\]
Slope from C to B is: \[\frac{ (y-(-2))}{ (x-3) }\]
and the slopes are related by the same perpendicular equation as above, so...
\[\frac{ (y-5)}{(x-4)} = \frac{-(x-3) }{(y+2) }\]

And the distance from A to B = dist from C to B, so use dist formula for both and equate them:
\[(x-4)^2 + (y-5)^2 = (x-3)^2 + (y+2)^2 \]

Answer 18

And from the slopes part above we have:
\[(y-5)(y+2)= -(x-3)(x-4)\] seems like there must be an easier way though...

Answer 19

Aaaah okay. Thank you :)

Answer 20

Is it making sense? It will work (i found a square yesterday with the same method) it just feels like it's more difficult than necessary.

Answer 21

Maybe it'll be easier once i expand out all the brackets. Some of the x^2 and y^2 terms cancel.

Answer 22

It's making some sense yes. I'll ask my teacher if all else fails.

Answer 23

Once you get one point via this difficult method, the other is easier (since you can then use the slopes of the lines)

Answer 24

K i think i may have found the y coordinates for both B and D...i'll check they make a square.

Answer 25

Okay. Thanks :D

Answer 26

lol, they didn't, but luckily i noticed a simple mistake and now have a square. Hopefully you can follow the working...

Answer 27

I basically just solved the equations I wrote a few posts above, and it gave both y coordinates for B and D.

Answer 28

Thank you!

Answer 29

You're welcome :)

Answer 30

@4meisu I googled a couple of things related to solving a square to see if there's an easier method but didn't find anything helpful.

I found this:
http://www.enotes.com/math/q-and-a/quadrilateral-abcd-where-4-5-c-3-2-square-find-383971

I'm guessing nmeisu is you, based on the name, and the fact the question was also posted today like this one... I can't see his answer though so idk what method he used.