(Photo credit: Alan Levine)
There has been a lot of feedback in the comments on my previous posts here demanding an article on Geometry. I have been avoiding writing it for some time because,
1. It is not one of my strong areas.
2. It takes a lot of time to draw the diagrams required to explain the fundas.
With the CAT having started, I realized that I could not procrastinate any more because Geometry is a very important part of CAT-prep. I have compiled a list of fundas that you might find helpful in solving CAT level questions. I am splitting those in two posts so that one post does not become too intimidating. In this first part, we will discuss Geometry fundas related to lines, triangles, parallelograms, trapeziums, polygons, etc.
You might find some of them very simple or obvious. If that is the case, be glad that your prep is up to the mark. If not, then be glad you learned them in time. (Yes I am inspired by two-face!)
Funda 1
The ratio of intercepts formed by a transversal intersecting three parallel lines is equal to the ratio of corresponding intercepts formed by any other transversal.
? a/b = c/d = e/f
Funda 2
The Centroid and Incenter will always lie inside the triangle. About the other points,
– For an acute angled triangle, the Circumcenter and the Orthocenter will lie inside the triangle.
– For an obtuse angled triangle, the Circumcenter and the Orthocenter will lie outside the triangle.
– For a right-angled triangle, the Circumcenter will lie at the midpoint of the hypotenuse and the Orthocenter will lie at the vertex at which the angle is 90.
Funda 3
The orthocenter, centroid, and circumcenter always lie on the same line known as Euler Line.
– The orthocenter is twice as far from the centroid as the circumcenter is.
– If the triangle is Isosceles then the incenter lies on the same line.
– If the triangle is equilateral, all four are the same point.
Funda 4: Appolonius Theorem {AD is the median}
AB2 + AC2 = 2 * (AD2 + BD2)
Funda 5: For cyclic quadrilaterals
Area = ?((s – a) (s – b) (s – c) (s – d)) where s is the semi perimeter s = (a + b + c + d)/2
Also, Sum of product of opposite sides = Product of diagonals
? ac + bd = PR * QS
Funda 6
If a circle can be inscribed in a quadrilateral, its area is given by = ?(abcd)
Funda 7: Parallelograms
- A parallelogram inscribed in a circle is always a Rectangle. A parallelogram circumscribed about a circle is always a Rhombus. So, a parallelogram that can be circumscribed about a circle and in which a circle can be inscribed will be a Square.
- Each diagonal divides a parallelogram in two triangles of equal area.
- Sum of squares of diagonals = Sum of squares of four sides,
AC2 + BD2 = AB2 + BC2 + CD2 + DA2
- A Rectangle is formed by intersection of the four angle bisectors of a parallelogram.
- From all quadrilaterals with a given area, the square has the least perimeter. For all quadrilaterals with a given perimeter, the square has the greatest area.
Funda 8: Trapeziums
- Sum of the squares of the length of the diagonals = Sum of squares of lateral sides + 2 Product of bases.
AC2 + BD2 = AD2 + BC2 + 2 x AB x CD
- If a trapezium is inscribed in a circle, it has to be an isosceles trapezium.
- If a circle can be inscribed in a trapezium, Sum of parallel sides = Sum of lateral sides.
Funda 9
- A regular hexagon can be considered as a combination of six equilateral triangles.
- All regular polygons can be considered as a combination of n isosceles triangles.
I will wrap up this post here. In my second post on Geometry we will discuss fundas related to circles (specifically, common tangents), solid figures, mensuration and co-ordinate geometry.
Ravi Handa, an alumnus of IIT Kharagpur, has been teaching for CAT and various other competitive exams for around a decade. He currently runs an online CAT coaching and CAT Preparation course on his website http://www.handakafunda.com