Hello Everyone,
It feels good to finally write an article on Math for PG. As the title correctly suggests, this is all about geometry. I do like Geometry but I am not one of those people who can balance vague 3-D structures all in the mind and come out with answers that go like “obviously the volume has to be 3500 cubic meters”. If you do wake me up in the night to ask about the ‘sine rule’, I will just ask for forgiveness for I have not sinned. So jokes apart, what is the purpose of this article?
Geometry is all about a lot of formulae, but it is difficult to remember them. What we will try to do is come up with some ways to understand and hence memorize formulae. We can also look at some general techniques that can help us solve a wide range of problems. To kick off the series we will start with Lines and Triangles followed by Polygons and Circles in the next part. Finally we shall look at Mensuration.
Parallel Lines
When you have a transversal cutting two parallel lines you can have two cases.
Case I: The transversal cuts the lines perpendicularly in this case all angles are equal to each other.
Case II: If the transversal does not cut perpendicularly then the angles formed are either acute or obtuse. Now all angles can be related in the following manner:
1. All acute angles are equal to each other
2. All obtuse angles are equal to each other
3. Any acute angle + any obtuse angle = 180
Triangle Inequality
Let’s say you have two sides ‘a’ and ‘b’. If you place them at an angle of 180 degrees, then you will get a straight line of length ‘a+b’, which will not lead to a triangle. To get a triangle you will have to nudge one of the sides up or down, hence the third side will always be less than the straight line ‘a+b’. In a similar fashion if you put one line on the top of another then the length between the two points is |a-b| again this will not be a triangle. To get a triangle you will need to nudge one of the sides again hence the third side will always be more than |a-b|. You do not need to check this with all combinations of sides, any two sides will do.
Interior and Exterior Angle Bisection Theorem
It is very easy to remember the interior angle bisection theorem because the ratio is of sides on the same side of the bisector. If you have a triangle ABC where angle A is bisected by AD then on one side of AD you have AB and BD on the other side you have AC and CD and guess what, their ratios are equal.
But unhappily on cursory observation, the exterior angle bisection theorem does not appear as beautiful. So in the same triangle if angle A is bisected externally and the angle bisector cuts line BC at X you will have AX lying outside the triangle. Now imagine this angle bisector actually pivoting inside and becoming the internal angle bisector then you will have AB/BX = AC/CX which is how it actually looks. So it is very easy to get the external bisection ratio by considering it to be an internal bisection.
Along with this you will need to know the Appollonius Theorem and the various methods for finding the area of a triangle.
Important Points
There are four different lines we can draw in a triangle, these lines are centered around a vertex and its bisection, perpendicularity and mid-point of opposite sides.
1. Vertices and mid-point of opposite sides give you medians.
2. Vertices and perpendicularity to the opposite sides give you altitudes.
3. Vertices and bisection of the angle at the vertex give you angle bisectors.
4. Perpendicular at the mid-point of sides give perpendicular bisectors.
5. Medians intersect to give us the centroid where centroid divides each median in the ratio 2:1.
6. Altitudes intersect to give you the orthocenter.
Now the major property of angle bisectors is that any point on the angle bisector is equidistant from the two sides of the vertex. So the intersection of the angle bisectors will be equidistant from the three sides. Hence you get the incenter which will be equidistant from the sides, that distance is the inradius and you can draw the incircle.
Similarly any point on the perpendicular bisector will be equidistant from the end-points of the segment. Hence the intersection will be equidistant from the vertices so that gives you the circumcenter and you can draw the circumcircle from the same point.
Important Triangles
Equilateral Triangle
Know all the following things inside out.
1. Area, height
2. It is composed of two 30-60-90 triangles
3. Circumradius = side / root(3)
4. Inradius = side / 2xsquareroot (3)
5. The only triangle where all points that is, incenter, circumcenter, orthocenter, centroid intersect.
Right angled triangle
Circumradius can be found as half of hypotenuse, Inradius can be found using the area formula i.e. 0.5xbxh = 0.5 (inradius)x(semi-perimeter).
Pay special attention to these triangles.
1. 3-4-5 know its inradius (1), its circumradius = 1/2 of hypotenuse = 2.5, its area.
2. 5-12-13 triangle, here inradius is 2, circumradius = 6.5.
3. 30-60-90 triangle can be seen as half of an equilateral triangle so sides opposite the respective angles bear the ratio of 1:squareroot(3):2.
4. 45-45-90 can be seen as half of a square so in a similar fashion sides bear the ratio of 1:1:squareroot(2).
How does knowing all these triangles help?
Good Question. A lot of times I have solved tougher triangle problems which was based on sides a,b,c and changed that to an equilateral triangle or if the sides are distinct to a 3-4-5 triangle got the required answer and re-substituted a,b,c with the required numbers. Believe me it is much faster. Also comfort with these basic triangles means a lot of on-paper solving reduces. So lets hope this was helpful. I will start looking squarely at polygons and circles now.
In case of doubts please send me a Private Message, as answering doubts on comments goes against the purpose of comments.
Munira Lokhandwala is an alumna from IIM Calcutta, batch of 1999. She has been associated with CAT coaching since 2001. In 2005, she started Catalyst Group tuitions for CAT. (www.catalyst4cat.com) she is a regular CAT taker herself. These are her scores:
Year – Overall percentile
2005 – 100 %ile
2004 – 99.99 %ile
2003 – 99.98 %ile