Without using distance formula, show that points $(-2,-1),(4,0),(3,3)$ and $(-3,2)$ are the vertices of a parallelogram.

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Geometry

NCERT Class 11 - Math

Chapter 10

Straight Lines

Section 1

Introduction

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Video Transcript

we have to prove that the 0.2 comma minus one. The points given to assad two comma sorry minus two comma minus 14 comma 03 comma three and minus three comma two is a parallelogram. These are the waters is of a hologram and you have to just prove this without using the distance formula. So as you can see where here the points have already been represented or in the figure itself. And we just have to prove, you have to prove that A. B. C. D. This is a Palo cramp. Right? So now how are we going to do this? It's already been specified that you do not have to use the distance formula. So without using the distance formula, what is the other way by which you can prove that this is a panda ground. So basically A B C. D. Is a parallelogram. If both pairs of opposing sides a panel to each other. That means if you just prove A B. Is powerless to be just too powerful to C. B. And the other side that is A. B. It is polished too busy. Then you can easily say that you are A B. C. D. Is a hologram. So that means over here we are going to prove that slope of A B is equal to slope off CB and slope of a B is equal to slope of BC. If you are able to prove these two things right, if you're able to prove these two steps, then you can easily conclude that your A. B. C. D. Is a Palo grande. So now we all know that the slope is given by the formula Y tu minus weber upon X two minus X one. So if you just absolutely values over hell. If I just talk about the Slope of the nine A. B. The slope of a B. That will be equal to y tu minus y one. If you just put the values of the waters is over here, the coordinates, then you'll be getting over here zero minus of minus one upon four minus of minus two. That means one upon four plus two, that is six. Similarly, if you talk about the slope of lying C. D then that is equal to slope of line C. D will be equal to 3 -2. up on three minus of minus three, that is plus three. So here, what that means one A 10.6 on simplification. So this is clear that slope of a B is equal to slope of CD. Right, so from here, since the slope is equal, that means a B is parallel to CD. Now we need to find in the slope of a B. And slope of BC. So now quickly just find out slope of a B. So slope of a B is equal to uh it's giving over here, you can just use the corners from hill, so slope of 80 will be equal to two plus one upon minus three plus two. So you can just substitute it over hill, as I say from the world is says, so over here will be happy two plus one, that is three upon minus three, close to that is minus one, so this is equal to minus three. And if we talk about the slope of busy, like a busy then slope of B C will be equal to on substituting the values over here, we will be getting three minus zero upon three minus four, this is equal to three upon minus one. Or you can say this is equal to minus three. So again the slope of 80 is equal to slope of Bc. Tack needs A. D. Is also a little busy. Now say this is one and this is to, we can see that oppose it size of talent to each other. So since the both pairs of opposite sides of a b c D, a parallel to each other, therefore, we can say that you're a b c D is a Palo grande, Hence proved. So as you can see, this was really very easy. In fact it was taking less time. If you could have, if you would have used in though that distance form light would have taken more time.

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