A HIGH SCHOOL MATH QUESTION..i really dont know how to do it..thankyoufor help...?

Question

Given quadrilateral PQRS with coodinates P(0,0)..Q(4,3)..R(7,-1)and S(3,-4)..
Prove that PQRS is a square....

What can i do with it..using slope?or someting?
I dont konw how to do the"Prove"with it...............= =

Answer 1

PQ = QR = RS = SP = 5

slope of PQ = 3/4
slope of QR = -4/3

Therefore PQ ⊥QR

Now, you have all four sides congruent and one angle is 90 degree. So you can say the quadrilateral must be a square.

Answer 2

I can get you started.

The characteristics of a square are:
It has 4 sides.
All 4 sides are equal in length.
The interior angles are all 90 degrees.

We are given that it is a quadrilateral, so it has 4 sides.

So, now you are left to prove the other two characteristics.
The easiest way is to show that each side (P-Q, Q-R, R-S, S-P) have the same total "rise" and "run".

P-Q: (0,0) to (4,3): 4,3
Q-R: (4,3) to (7,-1): 3,-4
R-S: (7,-1) to (3,-4): 4,-3
S-P: (3,-4) to (0, 0): -3,4

So, each side represents a shift of 3 in one direction and 4 in another. Since these values are the same, you know the lengths are the same. You can "prove" it by using Pythagoras' theorem to actually calculate the length of these sides:

(Length of side)^2 = (change in x)^2 + (change in y)^2

In all cases, the "Length of side" will be the same.

Finally, to prove that all the inside angles are right angles (90 degrees), you can use trigonometry. But that may be over complicating the problem. The fact that the rise and run are same on each side also indicates that the sides are at right angles to each other.

Answer 3

first thing is to graph those 4 points on a graph.
Use the definition of what a square is to say wether this is a square or not. Are all 4 sides equal? Do they form 90 degree angles?

You can use the formula for distance between 2 points to find out if this is true and that should be about it. If you find the 4 sides are indeed equal length then you may need to apply a little trigonometry to see if the Angles are all 90 degrees.

Answer 4

I think you could compute the length of the four sides, which will be equal.
Then, you can compute the length of the two diagonals, which will also be equal.
Once you have computed the lengths of the sides and diagonals, You can also use this and the pythagorean theorem to prove that the angles in each corner of the figure are 90-degree angles.

Answer 5

just prove that the four sides have the same length ...
then for just 1 angle, prove that it's 90 degree, could be by using the slope

Answer 6

Find distance PQ, QR, RS and SP.

in a square all sides are equal. PQ=QR=RS=SP
for square PR=SQ

Then find PR which is hypotenuse. and SQ

PQ=5=Sqrt(4^2+3^2)
QR=5=sqrt((7-4)^2+(-1-3)^2
RS=5=sqrt((3-7)^2+(-4+1)^2
SP=5=sqrt((3-0)^2+(-4-0)^2

PR=sqrt((7-0)^2+(-1-0)^2=sqrt(3-4)^2+(-4-3)^2)

Answer 7

determine the lengths of the sides, and you will see that they are equal. Use the pythagorean thereom.