Question

What type of quadrilateral is ABCD?

Answer 1

Either you go "a(b)?" or "a? (b)"
Don't go "a? (b)?"

Answer 2

I already calculated the slopes for each, but I'm having a bit of difficulty figuring out if one side is a negative reciprocal of another and likewise, or parallel/perpendicular. Here are the sides:
\[AB = \frac{ 6 }{ 9 }\]\[BC = \frac{ 3 }{ -2 }\]\[CD = \frac{ -6 }{ -9 }\]\[DA = \frac{ -3 }{ 2 }\]I can't exactly figure out if AB is perpendicular/a negative reciprocal to CD, and if the same goes for BC and DA. Because if they are parallel, they have no right angles, and therefore the shape cannot be a rectangle, right...? Thank you very much in advance!!

Answer 3

Simplify each of the fractions first? :)

Answer 4

Would that be dividing each of the fractions (i.e. -3 ÷ 2)? (I apologize if I'm incorrect...)

Answer 5

like change \(\dfrac69\) to \(\dfrac23\)

Answer 6

Ah, I see... so:
\[AB = \frac{ 2 }{ 3 }\]and \[CD = \frac{ -2 }{ -3 }\]...But would BC and DA (3/-2 and -3/2) change at all? I was about to say it would change to 1/1 or 1/2, but they both don't sound right to me...

Answer 7

ok here's what I would do:
\[\Large\begin{array}{ccccc}
m_{AB}&=&\frac69&=&\frac23\\
m_{BC}&=&\frac3{-2}&=&-\frac32\\
m_{CD}&=&\frac{-6}{-9}&=&\frac23\\
m_{DA}&=&\frac{-3}2&=&-\frac32
\end{array}\]

Answer 8

From that we can see that:
1. AB and CD are parallel
2. BC and DA are parallel
3. AB and BC are perpendicular
4. CD and DA are perpendicular

Answer 9

I see!! Thank you very much!! So since 2 sets of lines are parallel and the others perpendicular, I assume that would rule out it being a rectangle, and it would be a parallelogram...?

Answer 10

But you just said that it's a rectangle? :)

Answer 11

Oh..! I really apologize if I'm not clear since it's 3am here, haha..! I just assumed rectangles have all right angles, and that would only be possible if all sides were parallel -- or is that an incorrect description of it?

I'm not entirely sure if the quadrilateral having both parallel and perpendicular lines would contribute to it being a parallelogram more than a rectangle or likewise...

Answer 12

Correction:
From that we can see that:
1. AB and CD are parallel
2. BC and DA are parallel
3. AB and BC are perpendicular
4. BC and CD are perpendicular
5. CD and DA are perpendicular
6. DA and AB are perpendicular

Answer 13

Ah, that makes sense... man, thanks a ton for helping me out through all of this when it probably should have been more apparent to me a couple steps back! I really appreciate you being very thorough and helpful throughout this! Thank you very much for taking time to clear this up!!

Answer 14

no problem :)
Glad to know that I've helped you :D