# Pythagorean triplets

**Note**If each term of any pythagorean triplet is multiplied or divided by a constant (say P, P > 0) then the triplet so obtained will also be a pythagorean triplet. This is because if

*a*

^{2}+

*b*

^{2}=

*c*

^{2}, then (Pa)

^{2}+ (Pb)

^{2 }= (Pc)

^{2}, where P > 0.

3 Ã— 2 | 4 Ã— 2 | 5 Ã— 2 | gives |

6 | 8 | 10 | (6^{2 }+ 8^{2 }= 10^{2}) |

General formula for finding out all the primitive pythagorean triplets:

*a*=

*r*

^{2 }âˆ’

*s*,

^{2}*b*= 2

*rs*,

*c*=

*r*

^{2}+

*s*

^{2},

*r*>

*s*> 0 are whole numbers,

*r*âˆ’

*s*is odd, and

The greatest common divisor of

*r*and

*s*is 1.

# Important Triangle

**Orthocentre**

â€˜Oâ€™ is the orthocenter

*Property:*The point of intersection of the three altitudes of the triangle is known as the orthocenter. âˆ BOC = 190 âˆ’ âˆ A âˆ COA = 190 âˆ’ âˆ B

**Centroid**

â€˜Oâ€™ is the centroid

*Property:*The point of intersection of the three medians of a triangle is called the centroid. A centroid divides each median in the ratio 2 : 1 (vertex: base)

**Incentre**

â€˜Oâ€™ is the incentre

Property: The point of intersection of the angle bisectors of a triangle is known as the incentre. Incentre O is the always equidistant from all three sides i.e., the perÂpendicular distance between the sides.

**Circumcentre**

â€˜Oâ€™ is the incentre

*Property:*The point of intersection of the perpendicular bisectors of the sides of a triangle is called the circumcentre. OA = OB = OC = (circum radius) Circumcentre O is always equidistant from all the three vertices A, B and C Perpendicular bisectors need not be originating from the vertices.

# Important Theorems Related to Triangle

**45Â° â€“ 45Â° â€“ 90Â°**

âˆ A = 45Â° âˆ B = 90Â° âˆ C = 45Â°

*Explanation:*If the angles of a triangle are 45Â°, 45Â° and 90Â°, then thehypotenuse (i.e., longest side) is times of any smallerside. Excluding hypotenuse rest two sides are equal.

**30Â° â€“ 60Â° â€“ 90Â°**

âˆ C = 30Â°, âˆ B = 90Â°, âˆ A = 60Â°

Explanation: If the angles of a triangle are 30Â°, 60Â° and 90Â°, then the sides opposite to 30Â° angle is half of the hypotenuse and the side opposite to 60Â° is times the hypotenuse,

**Basic Proportionality Theorem (BPT)**

*Explanation:*Any line parallel to one side of a triangle divides the othertwo sides proportionally. So if DE is drawn parallel to BC, it would divide sides AB and AC proportionally i.e or .

**Mid-point Theorem**

*Explanation:*Any line joining the mid-points of two adjacent sides of a triangle are joined by a line segment, then this segment is parallel to the third side, i.e., if AD = BD and AE = CE then DE||BC.

**Apolloniusâ€™ Theorem**

*Explanation:*In a triangle, the sum of the squares of any two sides of a triangle is equal to twice the sum of the square of the median to the third side and square of half the third side. i.e., AB

^{2}+ AD

^{2}= 2 (AC

^{2}+ BC

^{2})

**Stewarts Theorem/Generalization of Apollonius Theorem**

*Explanation:*If length of AP = m and PB = n, then m Ã— CB

^{2}+ n Ã— AC

^{2 }= (m + n) PC

^{2}+ mn (m + n) Also understand that

*m*and

*n*here are length of segments, and not their ratio.

**Extension of Apolloniusâ€™ Theorem**

*Explanation:*In the given âˆ† ABC, AC, BE and DF are medians.

3 (Sum of squares of sides) = 4 (Sum of squares of medians) 3 (AB

^{2}+ AD^{2}+ DB^{2}) = 4 (AC^{2}+ EB^{2}+ FD^{2})**Interior Angle Bisector Theorem**

*Explanation:*In a triangle the angle bisector of an angle divides the opposite side to the angle in the ratio of the remaining twosides. i.e., and BD Ã— AC âˆ’ CD Ã— AB = AD

^{2}

^{}

^{}

**Exterior Angle Bisector Theorem**

*Explanation:*In a triangle the angle bisector of any exterior angle of a triangle divides the side opposite to the external angle in theratio of the remaining two sides i.e.,

# Congruency of Triangles

Rules for two triangles to be congruent

- S â€“ S â€“ S
- S â€“ A â€“ S
- A â€“ S â€“ A
- R â€“ H â€“ S

# Theorems for Similarity

- If in two triangles, the corresponding angles are equal, then their corresponding sides will also be proportional (i.e., in the same ratio). Thus the two triangles are similar.
- If the corresponding sides of two triangles are proÂportional (i.e., in the same ratio), their correspondÂing angles will also be equal and so the triangles are similar. This property is referred to as the SSS similarity criterion for the two triangles.
- If one angle of a triangle is equal to one angle of the other and the sides including these angles are proportional, then the triangles are similar. This property is referred to as the SAS similarity criterion of the two triangles.
- The ratio of the areas of the two similar triangles is equal to the ratio of the squares of their correÂsponding sides.
- If a perpendicular is drawn from the vertex of the right angle of a right angled triangle to the hypotenuse, the triangles on each side of the perpendicular are similar to the whole triangle and to each other.

# Quadrilaterals and Their Properties

- Area one diagonal Ã— (sum of perpendiculars to the diagonal from the opposite vertexes) d (h
_{1}_{ }+ h_{2}) - Area product of diagonals Ã— sine of the angle between them
- Area of the cyclic quadrilateral
*a*,*b*,*c*and*d*are the sides of quadrilateral and*s*= semiperimeter *Brahmaguptaâ€™s formula*For any quadrilateral with sides of length*a*,*b*,*c*and*d*, the area*A*is given by*a*and*d*, and B is the angle between the sides*b*and*c*.

# Circles and their Properties

**Centre**

*Definition:*The fixed point is called the centre. In the given diagram â€˜Oâ€™ is the centre of the circle.

**Radius**

*Definition:*The fixed distance is called a radius. In the given diagram OP is the radius of the circle. (point P lies on the circumference)

**Circumference**

*Definition:*The circumference of a circle is the distance around a circle, which is equal to 2Ï€r. (r â†’ radius of the circle)

**Secant**

*Definition:*A line segment which intersects the circle in two distinct points, is called as secant. In the given diagram secant PQ intersects circle at two points at A and B.

**Tangent**

(R is the point of contact) Note: Radius is always perpendicular to tangent.

*Definition:*A line segment which has one common point with the circumference of a circle, i.e., it touches only at only one point is called as tangent of circle. The common point is called as point of contact. In the given diagram, PQ is a tangent which touches the circle at a point R.

**Chord**

*Definition:*A line segment whose end points lie on the circle. In the given diagram AB is a chord.

**Diameter**

*Definition:*A chord which passes through the centre of the circle is called the diameter of the circle. The length of the diameter is twice the length of the radius. In the given diagram PQ is the diameter of the circle. (Oâ†’is the centre of the circle)

**Arc**

*Definition:*Any two points on the circle divides the circle into two parts the smaller part is called as minor arc and the larger part is called as major arc. It is denoted as â€˜â€™. In the given diagram PQ is arc.

**Semicircle**

*Defnation:*A diameter of the circle divides the circle into two equal parts. Each part is called a semicircle.

**Central Angle**

*Definition:*An angle formed at the centre of the circle, is called the central angle. In the given diagram âˆ AOB in the central angle.

**Inscribed Angle**

*Definition:*When two chords have one common end point, then the angle included between these two chords at the common point is called the inscribed angle. âˆ ABC is the inscribed angle by the arc ADC

**Measure of an Arc**

m(arc PRQ = m âˆ POQ

m(arc PSQ) = 360Â° âˆ’ m (arc PRQ)

m(arc PSQ) = 360Â° âˆ’ m (arc PRQ)

*Definition:*Basically, it is the central angle formed by an arc. e.g.,

(a) measure of a circle = 360Â°, (b) measure of a semicircle = 180Â°, (c) measure of a minor arc = âˆ POQ and (d) measure of a major arc = 360 âˆ’ âˆ POQ

**Intercepted Arc**

*Definition:*In the given diagram, AB and CD are the two intercepted arcs, intercepted by âˆ CPD. The end points of the arc must touch the arms of âˆ CPD, i.e., CP and DP.

**Concentric Circles**

*Definition:*Circles having the same centre at a plane are called the concentric circles.

_{1}and r

_{2}having the common (or same) centre. These are called as concentric circl Ces.

**Congruent Circles**

*Definition:*Circles with equal radii are called as congruent circles.

**Segment of a Circle**

*Definition:*A chord divides a circle into two regions. These two regions are called the segments of a circle: (a) major segment (b) minor segment.

**Cyclic Quadrilateral**

*Definition:*A quadrilateral whose all the four vertices lie on the circle.

**Circumcircle**

*Definition:*A circle which passes through all the three vertices of a triangle. Thus the circumcentre is always equidistant from the vertices of the triangle.

**In Circle**

Definition: A circle which touches all the three sides of a triangle i.e., all the three sides of a triangle are tangents to the circle is called an incircle. Incircle is always equidistant from the sides of a triangle.

# Summarizing the discussion regarding circle

1.Property: In a circle (or congruent circles) equal chords are made by equal arcs. {OP = OQ} = {Oâ€™R = Oâ€™S) PQ = RS and PQ = RS

2.

*Property:*Equal arcs (or chords) subtend equal angles at the centre PQ = AB (or PQ = AB) âˆ POQ = âˆ AOB

3.

*Property:*The perpendicular from the centre of a circle to a chord bisects the chord i.e., if OD âŠ¥ AB (OD is perpendicular to AB).

4.

*Property:*The line joining the centre of a circle to the mid-point of a chord is perpendicular to the chord. AD = DBOD âŠ¥ AB

5.

*Property:*Perpendicular bisector of a chord passes through the centre i.e., OD âŠ¥ AB and AD = DB âˆ´ O is the centre of the circle

6.

*Property:*Equal chords of a circle (or of congruent circles) are equidistant from the centre

7.

*Property:*Chords of a circle (or of congruent circles) are equidistant from the centre

8.

*Property:*The angle subtended by an arc (the degree measure of the arc) at the centre of a circle is twice the angle subtended by the arc at any point on the remaining part of the circle. m âˆ AOB = 2m âˆ ACB.

9.

*Property:*Angle in a semicircle is a right angle.

10.

*Property:*Angles in the same segment of a circle are equal i.e., âˆ ACB= âˆ ADB

11.

Property: If a line segment joining two points subtends equal angle at two other points lying on the same side of the line containing the segment, then the four points lie on the same circle. âˆ ACB = âˆ ADB
âˆ´ Points A, C, D, B are concyclic i.e., lie on the circle

12.

*Property:*The sum of pair of opposite angles of a cyclic quadrilateral is 180Â°.

13.

*Property:*Equal chords (or equal arcs) of a circle (or congruent circles) subtended equal angles at the centre.

14.

*Property:*If a side of a cyclic quadrilateral is produced, then the exterior angle is equal to the interior opposite angle. mâˆ CDE = m âˆ ABC

15.

Property: A tangent at any point of a circle is perpendicular to the radius through the point of contact.
(Inverse of this theorem is also true)

16.

*Property:*The lengths of two tangents drawn from an external point to a circle are equal i.e., AP = BP

17.

*Property:*If two chords AB and CD of a circle, intersect inside a circle (outside the circle when produced at a point E), then AE Ã— BE = CE Ã— DE

18.

*Property:*If PB be a secant which intersects the circle at A and B and PT be a tangent at T then PA Ã— PB = (PT)

^{2}

^{}

19.

^{}

*Property:*From an external point from which the tangents are drawn to the circle with centre O, then (a) they subtend equal angles at the centre (b) they are equally inclined to the line segment joining the centre of that point âˆ AOP = âˆ BOP and âˆ APO = âˆ BPO

20.

*Property:*If P is an external point from which the tangents to the circle with centre O touch it at A and B then OP is the perpendicular bisector of AB.

21.

*Property:*If from the point of contact of a tangent, a chord is drawn then the angles which the chord makes with the tangent line are equal respectively to the angles formed in the corresponding alternate segments. In the adjoining diagram.

âˆ BAT = âˆ BCA and âˆ BAP = âˆ BDA

22.

*Property:*The point of contact of two tangents lies on the straight line joining the two centres.

23.

*Property:*For the two circles with centre X and Y and radii r

_{1}and r

_{2}. AB and CD are two Direct Common Tangents (DCT), then the length of DCT

24.

*Property:*For the two circles with centre X and Y and radii r

_{1}and r

_{2}PQ and RS are two transverse common tangent, then length of TCT