Area and Collinearity

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Take the three points one comma two, three comma k, and five comma eight. Usually they make a triangle. But for exactly one value of k, there is no triangle at all. Which value is it, and why? Today one formula does two jobs: it gives the area of a triangle straight from its corners, and it tells us when three points lie on one line. This comes up often in JEE coordinate geometry, so let's begin.

0:34Area of a triangle from its vertices

So, here are the three vertices of a triangle: x one, y one; x two, y two; and x three, y three, each in round brackets. The small numbers below the letters are subscripts. They just number the points. Six numbers, and from them alone we want the area. Okay? Here is the formula. Delta, the Greek capital letter that looks like a little triangle, stands for the area. Delta equals one half, times the modulus of: x one times, in brackets, y two minus y three; plus x two times, y three minus y one; plus x three times, y one minus y two. The two straight bars around it are the modulus sign. Right? Now, don't learn this blindly. Look at the subscripts carefully, term by term. Do you see a pattern? Yes, exactly, they go round in a cycle! x one goes with two and three, x two with three and one, x three with one and two. One, two, three, and back to one; the arrows on screen show that cycle. Remember the cycle, and the formula writes itself. Got it? The same area can also be written with a determinant. Each row is one vertex: its x, its y, and a one. See the two pairs of bars: the inner straight bars make the determinant, and the outer bars are the modulus. Expand it down the first column, and you get exactly the bracket above. Clear? Now tell me. Why do we need the modulus around the bracket at all? Yes, exactly! The bracket can come out negative, for example when you list the vertices in the other order. But an area can never be negative. So always take the modulus, and never forget the half in front. With me?

2:26Collinear points: zero area

Now, think about our puzzle again. Imagine the third vertex sliding, slowly, until all three points sit on one straight line. What happens to the area of the triangle? Yes, exactly, the area becomes zero! The triangle is squashed flat. So three points are collinear, which means they lie on one line, if and only if this bracket is zero. The double arrow on screen means if and only if: it works both ways. Okay? And here is a small time saver. For collinearity, you need neither the half nor the modulus. Half the modulus is zero exactly when the bracket is zero. So just set the bracket equal to zero. Really simple! So, let us solve our puzzle. The points are one comma two, three comma k, and five comma eight. Use the cycle: one times, k minus eight; plus three times six, because eight minus two is six; plus five times, two minus k. Set it equal to zero. Pause the video and find k. Done? Okay, let's check. Opening the brackets: k minus eight, plus eighteen, plus ten, minus five k. That is twenty minus four k, equals zero. The arrow means implies, so k equals five. So the puzzle is solved: when k is five, there is no triangle, just a straight line. Right? And here is a quicker way. Look at the x coordinates. Three is exactly midway between one and five. So if the points are on one line, the middle point must be the mid-point, and k is two plus eight, over two, which is five. Same answer, much faster!

4:10Any polygon: the shoelace sum

Now, the same idea works for four corners, or five, or more. Write the vertices in order, going round the polygon: x one, y one, then x two, y two, all the way to x n, y n. The three dots mean and so on. In order means round the boundary, never jumping across. Okay? Then the area is one half, times the modulus of a sum. The big Greek letter sigma means add them all up. Here i is just a counter that runs over the vertices. Each term is x sub i times y sub i plus one, minus x sub i plus one times y sub i. In words: this x times the next y, minus the next x times this y. This is called the shoelace sum. Right? Now a small question. When the counter reaches the last vertex, which vertex is the next one? It goes back to the first vertex, yes! The polygon closes up, so the last vertex pairs with the first. Many students forget this last term. And with n equal to three, this sum gives exactly our triangle formula. Got it?

5:21Example 1: area of triangle (1, 2), (4, 6), (-2, 5)

Now, Example one, an easy one to warm up. Find the area of the triangle with vertices one comma two, four comma six, and minus two comma five. Pause the video and try it first. Done? Okay, let's check. First term: x one times, y two minus y three. That is one times, six minus five, which gives one. Okay? Second term: x two times, y three minus y one. Four times, five minus two, and that gives twelve. Right? Now the third term, and slowly here. x three times, y one minus y two, which is minus two, times, two minus six. What do you get? It is plus eight, very good! Two minus six is minus four, and minus two times minus four is plus eight. Many students write minus eight here, so watch the signs. So the bracket is one plus twelve plus eight, which is twenty one. Half the modulus of twenty one is twenty one over two, which is ten point five. And remember the half, because twenty one alone is wrong. Very good, you are doing great!

6:35Example 2: quadrilateral area by the shoelace sum

Now a harder one. Find the area of the quadrilateral with vertices zero comma zero, four comma zero, five comma three, and one comma four, in that order. Four corners, so which formula will you use? Pause the video and try it first. Done? Okay, let's check. Four corners, so it is the shoelace sum. First, vertex one with vertex two: x one times y two, minus x two times y one. Zero times zero, minus four times zero, which gives zero. The dot on screen just means times. Okay? Next, vertex two with vertex three: four times three, minus five times zero, which gives twelve. Right? Then vertex three with vertex four: five times four, minus one times three. Twenty minus three gives seventeen. Clear? So we have three terms. Shall we just add them up and finish? No, no, wait, there is one more term! The last vertex pairs with the first: one times zero, minus zero times four, which gives zero. Here it is zero, but often it is not, so never skip it. With me? So the area is one half, times the modulus of zero plus twelve plus seventeen plus zero. That is twenty nine over two, which is fourteen point five. Neat, right? Well done!

8:01Quick revision: area, collinearity, shoelace sum

So, quick revision. Area of a triangle: half the modulus of the bracket, with the subscripts going round in a cycle. Collinear points: set that same bracket equal to zero, no half, no modulus. And for any polygon, use the shoelace sum, with the vertices in order, and the last vertex joined back to the first. Now go and try the practice problems on this page. All the best!