7/28/2019 Quant Master Session - TGs Geometry II for CAT-12.pdf

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Quant Master Geometry Problems for CAT-12

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INRADIUS & CIRCUMRADIUS

Inradius (r) is radius of the incircle inscribed in a triangle which touches all the sides of the triangle.

Circumradius (R) is radius of circumcircle passing through all the vertices of triangle.

We have already mentioned the formulae to find r and R in terms of area of triangle.

*If b, c are two sides of triangle and h is the altitude drawn from vertex common to b and c, then R = bc/2h.

*For a right triangle with legs a, b and hypotenuse c, inradius r = (a + b c)/2

Congruent Triangles

The two triangles, whose any three corresponding parameters (out of three sides and three angles) are identical

are same to be congruent.

There are three exceptions to this rule and they are AAA, SSA and ASS.

Two congruent triangles are identical in all respects.

Similar Triangles

The two triangles are said to be similar (I) if two of the angles of triangles are same AA, or (II) if two sides of a

triangle are proportional to two sides of another triangle and angle included between the two sides in both the

triangles is same SAS.

Sides opposite to equal sides are said to be proportional sides and ratio of proportional sides is same.

Area of two similar triangles is in the ratio of square of ratio of corresponding sides.

7/28/2019 Quant Master Session - TGs Geometry II for CAT-12.pdf

2/2

Quant Master Geometry Problems for CAT-12

For any suggestions, corrections or improvements to this sheet and more of similar POWs, log on to our website

http://totagopinath.com

C. QUDRILATERALS

Area of a quadrilateral

If side lengths of a quadrilateral are a, b, c, d, then its area is not constant.

By changing internal angles, we can change area of the quadrilateral. For example take the case of a square withside length 5 whose area is 25 square units. Now keeping the side lengths constant, if we tilt it to turn it into a

rhombus, area goes o decreasing which can ultimately minimised to zero when the sides overlap with each other.

So, for a given set of four side lengths, Maximum Area is obtained for a cyclic quadrilateralwhich is given by

( )( )( )( )s a s b s c s d (BRAHMGUPTAs formula) where s is semiperimeter i.e. (a + b + c + d)/2.

Remember that for every four side lengths which form a quadrilateral, a cyclic quadrilateral can be formed.

If D1 and D2 are the two diagonals of a quadrilateral which intersects at angle P, then its area is given by

D1D2sinP.

And this area will be largest when diagonals intersect at right angle.

Ptolemys inequality

For a quadrilateral with side lengths a, b, c, d and diagonals of length D1 and D2, we have D1D2 ac + bd

Equality arises in case of a cyclic quadrilateral.

If two diagonals divide a quadrilateral in four triangles, as shown,

whose areas are A, B, C and D in order, then

A C = B D