Overview

Transcript

0:00Introduction

Hello everyone! First, a small challenge. Here is a triangle with sides five, seven and eight, and you have no protractor. Can you tell me one of its angles, exactly? Sounds impossible, right? By the end of this video, you will do it in one line. This chapter is about triangles, heights and distances, and its questions come up often in JEE papers. So, let's begin!

0:27The language of this chapter

So, here is triangle ABC, okay? The capital letters at the corners also name the angles there, and we will come to the two circles soon. Now, each side gets a small letter. Which side do you think is called small a? It is the side opposite the corner A, yes, exactly! Small a, b and c are the sides opposite the angles capital A, B and C, okay? So small a is the side B C, the side that does not touch the corner A at all. In the same way, small b is C A, and small c is A B, as the figure shows. Right? Next, add the three sides and divide by two. We call this s, the semi-perimeter: semi means half, so s is half the perimeter, not the full perimeter. Clear? And this capital Greek letter delta, the little triangle, stands for the area of the triangle. Every formula of the chapter is written in this language. Got it?

1:23Sine and cosine rules

Now, the two main rules of the chapter. First, the sine rule. Small a over sine A, equals small b over sine B, equals small c over sine C, and all of them equal two times capital R, okay? So every side, over the sine of its opposite angle, gives the same number. Right? Here, capital R is the circumradius: the radius of the grey circle through all three corners. Its centre is O, and the orange line is one radius. With me? Now suppose all three sides are known, but no angle at all. Which rule gives us an angle? It is the cosine rule, yes! Small a squared equals b squared plus c squared, minus two times b times c times cos A; the small two on top means squared. Okay? Rearrange it, and cos A equals b squared plus c squared minus a squared, over two times b times c, okay? Notice that the side opposite the angle takes the minus sign, and the same pattern gives the other two angles. Clear?

2:23Area and the two radii

Next, the area. When two sides and the angle between them are known, delta equals one half times b times c times sine A, okay? And when all three sides are known, delta equals the square root of s, times s minus a, times s minus b, times s minus c, all under the root sign. See, our semi-perimeter is already at work. Right? Now the circumradius again. From the sine rule, capital R equals small a, over two times sine A, and it also equals a times b times c, over four times delta. Got it? And small r, the inradius, is the radius of the green circle inside, with centre I, touching all three sides, and the short green line marked small r is one such radius. Small r equals delta over s, the semi-perimeter, okay? Keep that in mind for Example two.

3:12Heights and distances

The chapter ends with heights and distances. Here is a tower P Q of height h, with P the top and Q the foot. From the point A on level ground, we look up at the top, and the angle between this line of sight and the horizontal is the angle of elevation, alpha. From B, nearer the tower, it is beta, and the curly bracket marks d, the distance from A to B along the ground. Now stand at the top and look down at the point B, okay? The angle of depression is measured from which line? From the tower? No, not from the tower! It is always measured from the horizontal, the dashed line at the top. Looking down, the angle between the line of sight and the horizontal is the angle of depression. Clear? The dashed line is parallel to the ground, so these are alternate angles: the depression of B from P equals the elevation of P from B, which is beta. Got it? So, one habit: draw the figure first, mark every right angle, and write each tangent as opposite over adjacent in the correct right triangle. Most errors here come from the wrong triangle. With me?

4:22Example 1: angle B of a triangle with sides 5, 7, 8

Alright, remember our challenge? Here it is, from the section on the sine, cosine and projection rules. In triangle ABC, small a is five, small b is seven and small c is eight. Find the angle B, okay? Pause the video and try it first. Done? Okay, let's check. Three sides are known, so we use the cosine rule. Many students write cos B equals b squared plus c squared minus a squared, over two times b times c, from memory. Is that right? No, no, careful, that formula gives cos A! For the angle B, the opposite side, small b, takes the minus sign. So cos B equals a squared plus c squared minus b squared, over two times a times c, which is the correct formula here. Got it? Twenty five plus sixty four minus forty nine gives forty. Below, two times five times eight gives eighty; the dot means times. So cos B is forty over eighty, which is one half. Right? The angle whose cos is one half is sixty degrees; the small circle means degrees. So the angle B is exactly sixty degrees, in one line, with no protractor. So our challenge is solved!

5:38Example 2: a, area, R and r when A = 60 deg, b = 8, c = 5

Now a harder one, from the section on half angles, area and the radii. The angle A is sixty degrees, small b is eight and small c is five. Find small a, the area delta, capital R and small r, all four of them. Pause the video and try it first. Done? Okay, let's check. The cosine rule gives the third side. Two times eight times five times cos sixty degrees, which is one half, gives forty. So sixty four plus twenty five, minus forty, gives forty nine, and small a is seven. Okay? Next, the area: delta equals one half times b times c times sine A, as before. Small b times small c is forty, and sine sixty degrees is root three over two. So one half times forty times root three over two gives ten root three. Right? Now capital R equals small a, over two times sine A, from the sine rule. Two times root three over two is just root three, so capital R is seven over root three. Clear? And finally small r, which is delta over s, the semi-perimeter. The sides are seven, eight and five, and they add up to twenty. So s is twenty, isn't it? No, no, wait, that is the trap! Twenty is the full perimeter. The semi-perimeter s is half of it: seven plus eight plus five, over two, which is ten. Got it? So small r equals ten root three over ten, which is root three. So small a is seven, delta is ten root three, capital R is seven over root three, and small r is root three. Well done!

7:17Quick revision: cosine rule sign, area, R, r, elevation

So, quick revision. Small a, b and c are the sides opposite the angles capital A, B and C, and s is half the perimeter. In the cosine rule, the side opposite the angle takes the minus sign. Delta is one half times b times c times sine A, capital R is small a over two times sine A, and small r is delta over s, the semi-perimeter. And elevation and depression are always measured from the horizontal. Now work through this chapter and try its practice problems. All the best!