**Free Online Preparation for CAT with Minglebox e-CAT Prep. Cover basic concepts of Geometry under Quantitative Aptitude for MBA Entrance Exam Preparation with Study material, solved examples and tests prepared by CAT coaching experts. **

Two lines lying in a plane are said to be parallel if they never meet.

A closed figure, bounded by finite number of line segments, all lying in the same plane, is known as polygon.

Sum of interior angles of a polygon having 'n' sides, =

Sum of exterior angles of a polygon = 360 ( irrespective of number of sides)

A triangle is a polygon formed by 3 line segments joined end to end.

Sum of interior angles of a triangle = 180

In any given triangle sum of 2 interior angles is equal to the third remote angle.

AD is the median of triangle, then

G is the centroid of the triangle .Centroid of a triangle divides the median AD in the ratio 2:1

If AD is the bisector of the the angle ABC then AD divided the side BC in the ratio

ie

If AC is a chord of circle then angle ABC subtended by the chord at a point B on the circle is equal to

to angle subtended at point 0 lying in the same segment ie ∟ABC = ∟ADC. Also angle ABC is half of

angle AOC.

**Alternate Segment Theorem**

If XTB is a tangent to the circle then ∟ATB =∟ACT.

Area of a triangle =

=

where

Area of a circle of radius r =

Area of a square of side

Length of diagonal of square =

Area of rectangle of length l and breath b= lb

volume of cube of edge length

length of cube =

Volume of sphere of radius r =

Volume of hemisphere of radius r =

Surface area of cube =

Surface area of sphere =

Surface area of hemisphere =

**QUESTION**

The sum of the areas of two circles, which touch each other externally, is 153 . If the sum of their radii is 15, find the ratio of the larger to the smaller radius.

(1) 4

(2) 2

(3) 3

(4) none of these.

**SOLUTION**

Let the radii of the 2 circles be and , then = 15 (given)

and = (given)

= 153

solving we get, = 12 and = 3

ratio of the larger radius to the smaller one is 12 : 3 = 4 : 1 hence option (1) is the answer.

**QUESTION**

In the adjoining figure, points A, B, C and D lie on the circle. AD= 24 and BC = 12. what is the ratio of the area of CBE to that of the triangle ?

(1) 1:4

(2) 1:2

(3) 1:3

(4) data insufficient.

**SOLUTION**

In and , ∟ CBA = ∟ CDA.

( a chord of a circle subtends equal angel at all points on the circumference, lying in the same segment)

similarly ∟ BCD = ∟ BAD and ∟ BEC = ∟ AED

Therefore ( AAA similarity rule)

Now

Hence hence option b).

**QUESTION**

and are two concentric circles while BC and CD are tangents to at point P and respectively. AB to the circle , which of the following is true?

(1)

(2)

(3)

(4) data insufficient

**SOLUTION**

l ( cp) = l (cr) ( since tangents drawn from the same point are equal)

hence l(BC) = l( CD) ( points P and R are midpoints of BC and CD as OP is ┴BC and OR ┴ CD)

Therefore m( arc BXC) = m(arc CYD)

hence quadrilateral OPCR is a cyclic quadrilateral.

Hence answer is option c)

But m(arc BXC ) = ( ∟ABC) = ( tangent secant theorem)

therefore m(arc BZD) =

Therefore m(BCD) =

Hence m∟PCR =

m∟POR = 2m ∟PQR = (angle at center)

m∟OPC + m∟ORC =

Hence m∟POR + m∟PCR =

Therefore

**QUESTION**

Two circles of radius 3 units and 4 units are at some distance such that length of the transverse common tangents and the length of their direct common tangents are in the ratio 1: 2 . What is the distance between the centres of those circles?

(1) units

(2) units

(3) 8 units

(4) cannot be determined .

**SOLUTION**

Let x = distance between the centres of the circles.

T= length of the transverse common tangent.

D = length of direct common tangent.

And = 4 units and = 3 units

then, t = --------------- 1)

also d =

------------------- 2)

=

Since

units

hence

hence x =

from 1) and 2 ) we get

d = 2t

d = 8 units

hence answer is option 2 ).

amit39967 : need a bunch of 1000 questions

ankur36512 : i m not able to understand the concepts .. can any one help me to solve this..

shalu34890 : for deepak5949- height is same and base is double hence 1:2(example-2)

caroline13463 : concepts r short & crisp..really gud..but ans to second illustration is wrong

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saranya2768 : the ans. fr practice test Q.11 is 50

saranya2768 : in the practice test..ques no. 9 has error..i.e 516 is typed as 576

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i thank u all from the core of my heart

shri3834 : second problem answer is wrong.....

It is 1:4

area=1/2*a*b*sinC

so ratio consists of product of 2 sides...since angles are equal in both triangles.

so ratio areas is 2*2=4 i.e 1:4....

sonia4863 : give me a very good qns

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plz give me much more topic of DI,function mean QA,and passsage of english

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nitin481 : deepak5949 u r absolutely correct. In similar triangles ratio of area is equal to squares of ratio f their sides.

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deepak5949 : Solution to second illustration of geometry is incorrect.

The answer should be 1:4 and not 1:2 because ratio of area is required to be calculated.

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: material is good but tests are not workin ...it would be really helpful if the problem is sorted out asap

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