(Photo credit: Kai Schreiber)

This is my second post on PaGaLGuY focusing on Geometry fundas. We discussed lines, triangles, parallelograms, trapeziums and polygons in my previous post.

Now for some fundas and formulae in Geometry that happen to be frequently neglected by people but can fetch some crucial marks in the exam.

Funda 1: Angle made by Secants

? = 12 [ m(Arc AC) m(Arc BD) ]

? = 12 [ m(Arc AC) + m(Arc BD) ]

In both these cases, PA * PB = PC * PD

Funda 2: Common Tangents

Two Circles No. of Common Tangents Distance Between Centers (d) One is completely inside other 0 Touch internally 1 = r1 – r2 Intersect 2 r1 – r2 Touch externally 3 = r1 + r2 One is completely outside other 4 > r1 + r2

Length of the Direct Common Tangents (DCT)

? AD = BC = ?(d2 – (r1 – r2)2)

Length of the Transverse Common Tangent (TCT)

? RT = SU =?(d2 – (r1 + r2)2)

Note: The two centers(O and O), point of intersection of the DCTs (P)and point of intersection of TCTs (Q) are collinear. Q divides OO in the ratio r1 : r2 internally wheareas P divides OO in the ratio r1 : r2 externally.

Funda 3: Solids

– If a sphere is inscribed in a cube of side a, the radius of the sphere will be a/2. If a sphere is circumscribed about a cube of side a, the radius of the sphere will be ?3 * a/2

– If a largest possible sphere is inscribed in a cylinder of radius a and height h, its radius r will be

• r = h/2

  {If 2a > h}

• r = a

  {If 2a

– If a largest possible sphere is inscribed in a cone of radius r and slant height equal to 2r, then the radius of sphere = r /?3.

– If a cube is inscribed in a hemisphere of radius r, then the edge of the cube = r * ?(2/3).

Funda 4: On Coordinate Geometry

– The X axis divides the line joining P(x1,y1) and Q(x2,y2) in the ratio of y1 : y2

– The Y axis divides the line joining P(x1,y1) and Q(x2,y2) in the ratio of x1 : x2

– If we know three points A(x1,y1), B(x2,y2 ) and C(x3,y3) of a parallelogram, the fourth point is given by

(x1 + x3 x2, y1 + y3 y2)

With this I wrap up this post on Geometry. Best of Luck to all of you for CAT!

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