Distance and Section Formulae

Transcript

0:00Introduction

Hello everyone! Here is a challenge. Three consecutive corners of a parallelogram are one comma two, three comma four, and five comma one. Find the fourth corner. Most students draw a graph and solve two line equations, which takes minutes, right? By the end of this video, you will do it in ten seconds. Today we learn the distance and section formulae, which come up often in JEE exams. So, let's begin.

0:28Distance between two points

So, take two points: the point A is x one comma y one, and the point B is x two comma y two, in round brackets. The small one and two written low are subscripts, labels for the first and second point. Okay? How far apart are they? Go from A to B straight across, then straight up or down. Together with the segment A B, what shape do these two moves make? It is a right angled triangle, yes, exactly! The horizontal gap is x two minus x one, the vertical gap is y two minus y one, and A B is the hypotenuse. Right? So, by Pythagoras, A B is the square root of, x two minus x one, the whole squared, plus y two minus y one, the whole squared. The tick-shaped sign is the square root sign, and the small raised two means squared. Clear? Now one from the course: the distance between minus two comma three, and four comma minus five. The horizontal gap is four minus minus two. Quickly, is that two? Be careful, because four minus minus two is four plus two, which is six. The vertical gap is minus five minus three, which is minus eight. So A B is the square root of thirty six plus sixty four, which is ten. With me?

1:43Section formula: internal division

Now the second big idea. Take P between the two points. P divides A B internally in the ratio m is to n when A P is to P B equals m is to n. The colon is read as, is to. Clear? In the coordinates of P, the number m multiplies the coordinates of one point. Which one, A or B? It is B, and here is the formula. The x coordinate is m times x two, plus n times x one, over m plus n. The y coordinate is m times y two, plus n times y one, over m plus n. Right? So m multiplies the coordinates of B, and n those of A, a cross pairing. If m is bigger, A P is longer, so P sits closer to B, and B gets the bigger weight. Students often slip here, okay? The special case is the mid-point, ratio one is to one. It is x one plus x two over two, comma, y one plus y two over two. Got it?

2:41External division: replace n by −n

Now, a point can also divide a segment from outside: P on the line A B, beyond the segment, still with A P is to P B equal to m is to n. This is external division. With me? The formula changes in one way: replace n by minus n. So x is m times x two, minus n times x one, over m minus n, and the same for y. The equals sign with a slash means not equal: m cannot equal n, or the denominator is zero. Clear? So replacing n by minus n makes the ratio negative. That is why, if a ratio comes out negative, the division is external, okay?

3:20Mid-points and the parallelogram

Now, back to our puzzle. In a parallelogram ABCD, the diagonals A C and B D bisect each other, cutting each other in half. Right? So A C and B D have the same mid-point. Cancel the halves, and you get A plus C equals B plus D, adding x coordinates and y coordinates separately. Clear? Our consecutive corners are A at one comma two, B at three comma four, and C at five comma one. So D equals A plus C minus B, okay? But why subtract B, and not A or C? Because B is the middle corner, it is opposite to the fourth corner. Very good if you said that! So D is one plus five minus three, comma, two plus one minus four, which is three comma minus one. That took ten seconds, with no graph at all.

4:08Where the x-axis cuts AB

One last handy result. Suppose the x-axis cuts the line A B in the ratio m is to n. On the x-axis, the y coordinate is zero. Okay? So the y coordinate from the section formula is zero. The double arrow means, which gives: m times y two equals minus n times y one. With me? From this line, can you find the ratio m is to n yourself? Yes, exactly, the ratio m is to n equals minus y one, is to y two. Write this down. Got it? By our sign rule, a positive ratio means an internal cut, between the two points, and a negative ratio means an external cut. Clear?

4:49Example 1: internal and external division of (2,-3), (-4,6)

Alright, one from the course. Find the points dividing the join of two comma minus three, and minus four comma six, in the ratio two is to one, internally and externally. Pause the video and try it first. Done? Okay, let's check. The first point is A, the second is B, m is two and n is one. With me? Internal first, and remember, m goes with B, the second point. For x, two times minus four, plus one times two, over three. That is minus eight plus two, all over three, which gives minus two. For y, twelve minus three, all over three, gives three. So the internal point is minus two comma three. Right? Now externally, with n replaced by minus n. For y, the top is two times six, minus one times minus three. So is it twelve minus three again? Not quite, and this is the trap: minus one times minus three is plus three. So the top is twelve plus three, which is fifteen, and the bottom is just one. For x, minus eight minus two gives minus ten. So the external point is minus ten comma fifteen. Clear?

6:00Example 2: ratio in which the x-axis cuts (2, -3) to (5, 6)

Now a harder one from the course. In what ratio does the x-axis divide the join of two comma minus three, and five comma six, and at which point? Pause the video and try it first. Done? Okay, let's check. m is to n equals minus y one, is to y two. Here y one is minus three and y two is six, so the ratio is three is to six, which is one is to two. It is positive, so the cut is internal. Okay? Now the point. Many students write one times two, plus two times five, over three. Twelve over three gives four, so the point is four comma zero, right? No, no, wait, because m goes with the second point, five comma six. So x is one times five, plus two times two, over one plus two. Five plus four is nine, and nine over three gives three. Got it? Let's check y too: one times six, plus two times minus three, is zero, so the point is on the x-axis. So the ratio is one is to two, and the point is three comma zero.

7:06Quick revision: distance, section formula, parallelogram

So, quick revision. Distance is the square root of the x gap squared plus the y gap squared. In the section formula, m goes with B and n goes with A, and externally, replace n by minus n. In a parallelogram, A plus C equals B plus D, and the x-axis cuts the line A B in the ratio minus y one, is to y two. Now try the practice problems on this page. All the best!