## Pythagoras theorem and its application

"In a right angled triangle, **hypotenuse**^{2}** = base**^{2}** + altitude**^{2} "

**Pythagorean triplets**

^{2}= b

^{2}+ c

^{2}, then the triplet (a, b, c) of these three values is called a Pythagorean triplet.

8 : 15 : 17

**Special Pythagorean triplets**

In an isosceles right angled triangle, sides are in the ratio 1 : 1: √2

Exact half of an equilateral triangle is a right triangle with angles 30°, 60° and 90°. In such a triangle sides are in the ratio 1 : √3 : 2

**Interesting fact..!!!**

We can find out most of the Pythagorean Triplet by using the following general expressions;

**2m**, **m**^{2}** - 1** and **m**^{2}** + 1**, where m > 1.

## Polygons - a general approach

**Convex polygon:** In a polygon if all the interior angles are less than 180°, then it is a convex polygon. In a convex polygon, all the diagonals lie inside the polygon itself.

Concave polygon: If at least one of the angles of a polygon is a reflex angle, then the polygon is concave. At least one of its diagonals passes outside the polygon.

Almost all the B-school entrance exams limited the testing points only in convex polygons. Very rarely only exams like CAT and XAT testing the properties of concave polygons. As a general requirement, here we are considering convex polygons only.

**Name of polygons as per number of sides**

Number of sides (n) | Name |
---|---|

3 | Triangle |

4 | Equilateral |

5 | Pentagon |

6 | Hexagon |

7 | Septagon/Heptagon |

8 | Octagon |

9 | Nonagon |

10 | Decagon |

**Basic results:**

In any 'n' sided polygon;

- Sum of interior angles = (n - 2) 180°
- Sum of exterior angles = 360°
- Number of diagonals from any particular vertex to rest all vertices's = n - 3
- Total number of diagonals = n/( n-3)2
- Number of triangles formed by joining vertices's = n ( n-1)(n-2)/6

**Regular polygons:**

In a polygon if all sides are equal in length and all the angles are equal in measure, then it is a regular polygon.

Eg: square, equilateral triangle etc.

**GEOMETRY & MENSURATION**

In this section, we are mainly discussing about the basic properties of various quadrilaterals and circles.

## Quadrilaterals

**Trapezium**

One pair of opposite sides are parallel.

i.e. AB || CD.

Height of a trapezium is equal to the distance between the parallel sides.

Area = 1/2 * h(a+b), where a and b are the lengths of the parallel sides and h is distance between the parallel sides.

In a trapezium if the lengths of the non-parallel sides are equal, then it is an isosceles trapezium.

**Parallelogram**

Pair of opposite sides are parallel. i.e. AB || CD and BC || AD .

Height of a parallelogram is the distance between two parallel sides.

Diagonals bisect each other. i.e. M is the midpoint of AC and BD.

Δ AMB ≅ Δ CMD and ΔAMD ≅ ΔCMB.

Δ AMB, Δ CMD, ΔAMD and ΔCMB are all equal in area. i.e. Area of each triangle =

1/4 * Area of the parallelogram

Area = b * h, where 'b' is base and 'h' is height.

**Rhombus**

All sides are equal

Diagonals bisect each other at 90°.

ΔAMB ≅ ΔCMD ≅ ΔAMD ≅ ΔCMB. area of each triangle = 1/4 * Area of the rhombus

Area = 1/2 * (d1 * d2), where d_{1} and d_{2} are the diagonals.

**Rectangle**

Opposite sides are equal in length.

Opposite sides are parallel.

Diagonals are equal and bisect each other.

Four triangles formed by the intersection of two diagonals are all equal in area. i.e. area of each triangle =

1/4 * Area of the rectangle.

Out of the above mentioned triangles, opposite triangles are congruent.

Area = length * breadth

Perimeter = 2 ( length + breadth)

- Square

All sides are equal.

All angles are right angles.

Diagonals are equal and bisect each other at 90°.

All the four triangles formed by the intersection of diagonals are congruent.

Area = side^{2} = a^{2}

Perimeter = 4a

Length of diagonal = √2 * a

## Circles

Circle is the collection of all points which are equidistant from a particular point on a plane. This particular point is called the center of the circle and constant distance is the radius of the circle.

O - center

r - radius

Diameter(d) = 2r

Area = π r2

Circumference = 2 πr

Angle around the centre = 360°

**Area and arc length of a sector:**

If the central angle of a sector = θ°, then;

θ/360 = arc length of the sector/ 2 πr = area of the sector/ π r2

Area of sector = θ3/60 * π r2

Arc length of the sector = θ/360 * 2πr

**Rules:**

- Perpendicular from the center of a circle to a chord will bisect the chord.
- If two chords are intersecting in a circle, then;

PT * QT = RT * ST

- If any two secants of a circle are intersecting then;

CA * CB = CE * CD

- Angle inscribed in a semi circle is right angle.

**Central angle property:**

Angle subtended at the center of a circle by an arc is double the angle subtended by it at any point on the remaining part of the circumference.

Angles in the same segment of a circle are equal.

OR

Angles lie in the same arc are equal.

**Tangents and secants to a circle**

If a line touches the circle at exactly one point, then the line is a tangent to the circle.

- AB is a tangent and T is the point of tangency.
- There are infinitely many tangents are possible to a circle.
- Through one point on the circumference of a circle, there is exactly one tangent is possible to a circle.
- From an outside point, there are two different tangents are possible to a circle.

In the diagram PQ and PR are the tangents from an outside point P.

PQ = PR

Radius to the point of tangency is perpendicular to the tangent.

- In the below diagram AB is a tangent and ACD is a secant to the same circle.

AB^{2} = AC * AD

- In the below diagram PBA and PDC are two secants to the same circle.

PB * PA = PC * PD

**Alternate segment theorem:**

The angle between a tangent and a chord through the point of contact of the tangent and is equal to the angle made by the chord in the alternate segment.

∠PQR = ∠QSR

**Cyclic quadrilateral**

A quadrilateral inscribed in a circle is called Cyclic Quadrilateral.

In diagram ABCD is a cyclic quadrilateral.

Area of a cyclic quadrilateral = √((S-a)(S-b)(S-c)(S-d))

Where a, b, c and d are the four sides of the quadrilateral and S = a+b+c+d / 2

Opposite angles of a cyclic quadrilateral are supplementary. i.e. ∠A+∠C= ∠B+ ∠D=180°

**Ptolemy's Theorem:**

If ABCD is a cyclic quadrilateral, then the product of the two diagonals is equal to the sum of the product of the opposite sides.

AC * BD = (AB * CD) + (AD * BC)

In this section on Geometry and Mensuration, we are dealing with 'Three Dimensional Geometry'. The term Three Dimensional Geometry represents a vast area of mathematical application, even in higher mathematical concepts too. However our requirement is as simple as to deal with some of the important concepts about Solid Geometry. The section 'Solid Geometry' mainly contains the concepts of Surface Areas and Volumes of various solid figures. Again there is a limitation in this regard; here we are dealing only with some particular type of solid figures, such as Prisms, Pyramids and Spheres. Most of the questions from this section of Geometry in MBA entrance exams are of application type, neither theoretical nor property related.

- Prisms
- Right Prisms
- Classification of Prisms
- Properties and results related to prisms
- Pyramids
- General properties and results of Pyramids
- Cone and Frustum of a cone
- Sphere and Hemi-sphere