Vector algebra (proving you have a parallelogram by using vectors)

In summary: BP.)It is not enough, you have to point out which sides in ABD and in BCD correspond to each other, and these sides are parallel.DR an PB must be parallel since PD and BR are parallel and have the same length BS and QD are also parallelAD also has the same length as BCand AB has the same length as CDSo AB=CD and AD=BCTherefore AB must be parallel to CD and AD must be parallel to BC
  • #1
Matejxx1
72
1

Homework Statement


23. In a ABCD quadrilateral let P,Q,R,S be midpoints of sides AB,BC,CD and DA. Let X be the intersection of BR and DQ, and let Y be the intersection of BS and DP. If ##\vec{BX}=\vec{YD} ## show that ABCD is a parallelogram .

Slika nove bitne slike.jpg

Homework Equations


## (\vec{a}\cdot\vec{b})=0## then a and b are perpendicular
I can't think of any other useful ones

The Attempt at a Solution


20161106_230723.jpg
[/B]
So when I first saw this problem I didn't really have an idea how to solve it so I just tried figuring out what are the rations that X divides BR into. I calculated that the ratio is 2/3. I did the same for Y and DP and also got the ratios 2/3. Which I though was good since I figured out that ##\vec{BX}=\vec{YD} ## really are the same. However I still didn't have any idea how to continue so then I decided to figure out ratios of AY,AX,XY and found out that AY/(AC)=XY/(AC)=XC/(AC)=1/3 however I don't see how any of this would help me show that ABCD is a parallelogram. I'm kinda lost here.
I though about trying to show that ## (\vec{a}\cdot\vec{b})=/=0## since that would mean that a and b are not perpendicular and that we have a parallelogram however I was not able to find a way to show that.
Any help is greatly appreciated.
 
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  • #2
I'd have started with a diagram of a random quadralateral ... rather than a rectangle.
You need to keep in mind the definition of a parallelogram ... play about with the diagrams, look at how the geometry differs for different parralellograms.
 
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  • #3
Matejxx1 said:
So when I first saw this problem I didn't really have an idea how to solve it so I just tried figuring out what are the rations that X divides BR into. I calculated that the ratio is 2/3. I did the same for Y and DP and also got the ratios 2/3. Which I though was good since I figured out that ##\vec{BX}=\vec{YD} ## really are the same. However I still didn't have any idea how to continue so then I decided to figure out ratios of AY,AX,XY and found out that AY/(AC)=XY/(AC)=XC/(AC)=1/3 however I don't see how any of this would help me show that ABCD is a parallelogram. I'm kinda lost here.
.

By definition, BR and DQ are medians in the green triangle BCD, X is the center of gravity and it divides the medians in the ratio 1/2. It is the same for Y in triangle ABD. The thick blue and red vectors are of the same size and have the same angle with the BD, the common side of the two triangles. Find congruent triangles.

upload_2016-11-11_5-37-23.png
 
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  • #4
Sorry for taking so long to reply was kinda busy with some school work

ehild said:
By definition, BR and DQ are medians in the green triangle BCD, X is the center of gravity and it divides the medians in the ratio 1/2. It is the same for Y in triangle ABD. The thick blue and red vectors are of the same size and have the same angle with the BD, the common side of the two triangles. Find congruent triangles.

View attachment 108748
so congruent triangles are:
BXD and BYD
DPB and DRB
QDB and SBD
SPY and RQX
ASP and QCR
so that must mean that
ABD and BCD are also congruent
 
  • #5
Matejxx1 said:
Sorry for taking so long to reply was kinda busy with some school workso congruent triangles are:
BXD and BYD
DPB and DRB

What about the angles? Are DR and PB parallel?
Matejxx1 said:
QDB and SBD
SPY and RQX
ASP and QCR
so that must mean that
ABD and BCD are also congruent

It is not enough, you have to point out which sides in ABD and in BCD correspond to each other, and these sides are parallel.
 
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  • #6
DR an PB must be parallel since PD and BR are parallel and have the same length
BS and QD are also parallel
AD also has the same length as BC
and AB has the same length as CD
So AB=CD and AD=BC
Therefore AB must be parallel to CD and AD must be parallel to BC
 
  • #7
Matejxx1 said:
DR an PB must be parallel since PD and BR are parallel and have the same length
BS and QD are also parallel
AD also has the same length as BC
and AB has the same length as CD
So AB=CD and AD=BC
Therefore AB must be parallel to CD and AD must be parallel to BC
It is all right! But you should have said why the triangles are congruent (for example, BXD and BYD have two sides DY and BX equal, one side (BD) common, and the same angle between them. And it would have been enough to show that DR || PB involving DC || AB and SD || BQ, involving AD || BC.
 
  • #8
Since this problem has been solved, I will show you a vector proof. I will start by assuming you know the medians intersect ##\frac 2 3## the way across a triangle. Then, using the same picture (reproduced here):
upload_2016-11-11_5-37-23-png.108748.png

Note that ##BX = \frac 2 3(BC + \frac 1 2 CD)## and ##YD = \frac 2 3(AD - \frac 1 2 AB)##. You are given that ##BX = YD## which implies ##BC + \frac 1 2 CD=AD - \frac 1 2 AB## or ##BC - AD = \frac 1 2 (-CD - AB)##. These are vectors, so we can write this as ##BC + DA = \frac 1 2 (DC + BA)##.
But from the given quadrilateral we have ##AB + BC + CD + DA = \vec 0##. From this we get ##BC + DA = -AB -CD = BA + DC##. Substituting this in the left side of the last equation in the previous paragraph gives ##BA+DC =\frac 1 2(DC+BA)##, which implies ##DC+BA = \vec 0## or ##AB = DC##. So these opposite sides are equal and parallel.
 

1. How do you prove that a quadrilateral is a parallelogram using vector algebra?

To prove that a quadrilateral is a parallelogram using vector algebra, you must show that opposite sides are parallel and equal in length. This can be done by calculating the vectors of each side and using properties of vector addition and subtraction to show that they are parallel and equal in magnitude.

2. What is the formula for calculating the magnitude of a vector?

The formula for calculating the magnitude (or length) of a vector is |v| = √(x² + y²), where x and y are the components of the vector in the x and y directions respectively.

3. Can you use vector algebra to prove that a rectangle is a parallelogram?

Yes, you can use vector algebra to prove that a rectangle is a parallelogram. Since a rectangle has all sides equal and all angles equal to 90 degrees, you can show that the opposite sides are parallel and equal in length using the same method as proving a general quadrilateral is a parallelogram.

4. What is the difference between a vector and a scalar?

A vector is a mathematical object that has both magnitude and direction, whereas a scalar only has magnitude. In the context of vector algebra, we use vectors to represent physical quantities that have both size and direction, such as displacement, velocity, and force.

5. Is vector algebra commonly used in real-world applications?

Yes, vector algebra is widely used in various fields of science and engineering, such as physics, computer graphics, and navigation. It allows us to represent and manipulate physical quantities with both magnitude and direction, making it a powerful tool for solving real-world problems.

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