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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 a2 + b2 = c2, then (Pa)2 + (Pb)2 = (Pc)2, where P > 0.
 
For example,
3 × 2 4 × 2 5 × 2 gives
6 8 10 (62 + 82 = 102)

General formula for finding out all the primitive pythagorean triplets:
a = r2 − s2,
b = 2 rs,
c = r2 + s2,
r > s > 0 are whole numbers,
r − s is odd, and
The greatest common divisor of r and s is 1.

Important Triangle

Orthocentre
 
Description: Description: 23086.png
‘O’ is the orthocenter

Property: The point of intersection of the three altitudes of the triangle is known as the orthocenter. BOC = 190 − A COA = 190 −  B
 
AOB = 180 −  B
 
 

Centroid
 
Description: Description: 23093.png
‘O’ 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) Description: Description: 6818.png
 
 

Incentre
Description: Description: 23100.png
‘O’ 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 per­pendicular distance between the sides.
 
 

Circumcentre
Description: Description: 23107.png
‘O’ 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

45° – 45° – 90°
 
Description: Description: P-330-1.tif
∠A = 45° ∠B = 90° ∠C = 45°

Explanation: If the angles of a triangle are 45°, 45° and 90°, then thehypotenuse (i.e., longest side) is Description: Description: 6768.png times of any smallerside. Excluding hypotenuse rest two sides are equal.
 
i.e., AB = BC and AC Description: Description: 6761.pngAB Description: Description: 6754.png BC
 
AB : BC : AC = 1 : 1 : Description: Description: 27974.png
 
 

30° – 60° – 90°
 
Description: Description: P-330-2.tif
∠C = 30°, ∠B = 90°, ∠A = 60°

Explanation: If the angles of a triangle are 30°, 60° and 90°, then the sides opposite to 30° angle is half of the hypotenuse and the side opposite to 60° is Description: Description: 6739.png times the hypotenuse,
 
e.g., AB Description: Description: 6732.png and Description: Description: 6725.png AC
 
AB : BC: AC = 1: Description: Description: 6718.png
 
 

Basic Proportionality Theorem (BPT)
 
Description: Description: P-330-3.tif
 
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 Description: Description: 11251.png or Description: Description: 11244.pngDescription: Description: 11237.png.
 

Mid-point Theorem
 
Description: Description: P-330-4.tif
 
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.
 

Apollonius’ Theorem
 
Description: Description: P-330-5.tif
 
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., AB2 + AD2 = 2 (AC2 + BC2)
 

Stewarts Theorem/Generalization of Apollonius Theorem
 
Description: Description: P-330-6.tif
 
Explanation: If length of AP = m and PB = n, then m × CB2 + n × AC2 = (m + n) PC2 + mn (m + n) Also understand that m and n here are length of segments, and not their ratio.
 
 

Extension of Apollonius’ Theorem
 
 
Description: Description: 23132.png

Explanation: In the given ∆ ABC, AC, BE and DF are medians.
 
3 (Sum of squares of sides) = 4 (Sum of squares of medians) 3 (AB2 + AD2 + DB2) = 4 (AC2 + EB2 + FD2)
 
 

Interior Angle Bisector Theorem
 
Description: Description: 23147.png
 
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., Description: Description: 6628.png and BD × AC − CD × AB = AD2
 

Exterior Angle Bisector Theorem
 
Description: Description: 23176.png
 
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., Description: Description: 6613.png

Congruency of Triangles

Rules for two triangles to be congruent
  1. S – S – S
     
    If in any two triangles, each side of one triangle is equal to a side of the other triangle, the two triangles are congruent. This rule is S – S – S rule.
  2. S – A – S
     
    In  ABC and  ABD, AB = AB (common side)
     
    ∠ ABC = ∠ BAD (given)
     
    BC = AD (given)
     
    Description: Description: 11349.png
     
    Thus by rule S – A – S the two triangles are congruent.
     
    This rule holds true, when the angles that are equal have to be included between the two equal sides
     
    (i.e., the angle should be formed between the two sides that are equal). 
  3. A – S – A
     
    In  ABC and  ADE,
     
    ∠ ACB = ∠ AED (given)
     
    ∠ BAC = ∠ DAE (common angle)
     
    BC = DE (given)
     
    Thus by rule A – S – A the two triangles are congruent.
     
    For this rule, the side need not be the included side.
     
    Description: Description: 6581.png
     
    A – S – A can be written as A – A – S or S – A – A also.
  4. R – H – S
     
    This rule is applicable only for right-angled triangles.
     
    If two right-angled triangles have their hypotenuse and one of the sides as same, then the triangles will be congruent.

Theorems for Similarity

  1. 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.
     
    This property is referred to as the AAA similarity criterion for two triangles.
     
    Corollary If two angles of a triangle are respec­tively equal to two angles of another triangle, then the two triangles are similar. This is referred to as the AA similarity criterion for the two triangles. It is true due to the fact that if two angles of one triangle are equal to the two angles of another tri­angle, then the third angle of both the triangles will automatically be the same.
  2. If the corresponding sides of two triangles are pro­portional (i.e., in the same ratio), their correspond­ing 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.
  3. 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.
  4. The ratio of the areas of the two similar triangles is equal to the ratio of the squares of their corre­sponding sides.
  5. 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

  1. Area Description: Description: 6441.png one diagonal × (sum of perpendiculars to the diagonal from the opposite vertexes) Description: Description: 6433.png d (h1 + h2)
  2. Area Description: Description: 6426.png product of diagonals × sine of the angle between them
  3. Area of the cyclic quadrilateral
     
    Description: Description: 6419.png where abc and d are the sides of quadrilateral and s = semiperimeter
     
    Description: Description: 6412.png
  4. Brahmagupta’s formula For any quadrilateral with sides of length abc and d, the area A is given by
     
    Description: Description: 6405.png
     
    Where Description: Description: 6398.png is known as the semi-perimeter, A is the angle between sides a and d, and B is the angle between the sides b and c.

Circles and their Properties

Centre
 
Description: P-338-1.tif
 
Definition: The fixed point is called the centre. In the given diagram ‘O’ is the centre of the circle.
 
 

Radius
 
Description: P-338-2.tif
 
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
 
Description: P-338-3.tif
 
Definition: The circumference of a circle is the distance around a circle, which is equal to 2πr. (r → radius of the circle)
 
 

Secant
 
Description: P-338-4.tif
 
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
 
Description: P-338-5.tif
(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
 
Description: P-338-6.tif
 
Definition: A line segment whose end points lie on the circle. In the given diagram AB is a chord.
 
 

Diameter
 
 
Description: P-338-7.tif
 
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. (Ois the centre of the circle)
 
 

Arc
 
Description: P-338-8.tif
 
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 ‘’. In the given diagram PQ is arc.
 
 

Semicircle
 
Description: P-339-1.tif
 
Defnation: A diameter of the circle divides the circle into two equal parts. Each part is called a semicircle.
 
 

Central Angle
 
Description: P-339-2.tif
 
Definition: An angle formed at the centre of the circle, is called the central angle. In the given diagram AOB in the central angle.
 
 

Inscribed Angle
 
 
Description: P-339-3.tif
 
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. ABC is the inscribed angle by the arc ADC
 
 

Measure of an Arc
 
Description: P-339-4.tif
m(arc PRQ = m POQ
m(arc PSQ) = 360° − m (arc PRQ)

 
Definition: Basically, it is the central angle formed by an arc. e.g.,
(a) 
measure of a circle = 360°, (b) measure of a semicircle = 180°, (c) measure of a minor arc = POQ and (d) measure of a major arc = 360 − POQ

 

Intercepted Arc
 
Description: 23275.png

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

Concentric Circles
 
Description: 23267.png
 
Definition: Circles having the same centre at a plane are called the concentric circles. 
 
In the given diagram, there are two circles with radii r1and r2 having the common (or same) centre. These are called as concentric circl Ces.
 
 

Congruent Circles
 
Description: P-339-7.tif
 
Definition: Circles with equal radii are called as congruent circles.
 
 

Segment of a Circle
 
Description: P-340-1.tif
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
 
Description: P-340-2.tif
 
Definition: A quadrilateral whose all the four vertices lie on the circle.
 
 

Circumcircle
 
Description: P-340-3.tif
 
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.
 
OA = OB = OC (circumradius)
 
 

In Circle
Description: P-340-4.tif
 
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.
Description: Description: P-346-1.tif

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

2.
Description: Description: P-346-2.tif
 
Property: Equal arcs (or chords) subtend equal angles at the centre PQ = AB (or PQ = AB) ∠POQ = ∠AOB
 
 
 
3.
Description: Description: P-346-3.tif
 
Property: The perpendicular from the centre of a circle to a chord bisects the chord i.e., if OD ⊥ AB (OD is perpendicular to AB).
 
 

4.
Description: Description: P-346-4.tif
 
Property: The line joining the centre of a circle to the mid-point of a chord is perpendicular to the chord. AD = DBOD ⊥ AB
 
 

5.
Description: Description: P-346-5.tif
 
Property: Perpendicular bisector of a chord passes through the centre i.e., OD ⊥ AB and AD = DB ∴ O is the centre of the circle
 
 

6.
Description: Description: P-346-6.tif
 
Property: Equal chords of a circle (or of congruent circles) are equidistant from the centre
 
∴ AB = PQ
 
∴ OD = OR
 
 

7.
Description: Description: P-346-7.tif
 
Property: Chords of a circle (or of congruent circles) are equidistant from the centre
 
∴ OD = OR
 
∴ AB = PQ
 
 

8.
Description: Description: P-347-1.tif
 
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 ∠AOB = 2m ∠ACB.
 
 

9.
Description: Description: P-347-2.tif
 
Property: Angle in a semicircle is a right angle.
 
 

10.
Description: Description: P-347-3.tif
 
Property: Angles in the same segment of a circle are equal i.e., ∠ACB= ∠ADB
 
 

11.
Description: Description: P-347-4.tif
 
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. ∠ACB = ∠ADB
 
∴ Points A, C, D, B are concyclic i.e., lie on the circle
 
 

12.
Description: Description: P-347-5.tif
 
Property: The sum of pair of opposite angles of a cyclic quadrilateral is 180°.
 
∠DAB + ∠BCD = 180° and ∠ABC + ∠CDA = 180° (Inverse of this theorem is also true)
 
 

13.
Description: Description: P-347-6.tif
 
Property: Equal chords (or equal arcs) of a circle (or congruent circles) subtended equal angles at the centre.
 
AB = CD (or AB = CD) ∠AOB = ∠COD
 
(Inverse of this theorem is also true)
 
 

14.
Description: Description: P-347-7.tif
 
Property: If a side of a cyclic quadrilateral is produced, then the exterior angle is equal to the interior opposite angle. m∠CDE = m ∠ABC
 
 

15.
Description: Description: P-348-1.tif
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.
Description: Description: P-348-2.tif
 
Property: The lengths of two tangents drawn from an external point to a circle are equal i.e., AP = BP
 
 

17.
Description: Description: P-348-3.tif
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 × BE = CE × DE
 
 

18.
Description: Description: P-348-4.tif
 
Property: If PB be a secant which intersects the circle at A and B and PT be a tangent at T then PA × PB = (PT)2
 
 

19.
Description: Description: P-348-5.tif
 
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 ∠AOP = ∠BOP and ∠APO = ∠BPO
 
 

20.
Description: Description: P-348-6.tif
 
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.
 
OP ⊥ AB and AC = BC
 
 

21.
Description: Description: P-348-7.tif
 
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.
∠BAT = ∠BCA and ∠BAP = ∠BDA
 
 

22.
Description: Description: P-349-1.tif
Property: The point of contact of two tangents lies on the straight line joining the two centres.
 
(a) When two circles touch externally then the distance between their centres is equal to sum of their radii, i.e., AB = AC + BC
 
(b) When two circles touch internally the distance between their centres is equal to the difference between their radii. i.e., AB = AC – BC
 
 
 
23.
Description: Description: P-349-2.tif

Property: For the two circles with centre X and Y and radii r1 and r2. AB and CD are two Direct Common Tangents (DCT), then the length of DCT Description: Description: 5305.png
 
 

24.
Description: Description: P-349-3.tif
Property: For the two circles with centre X and Y and radii r1 and r2 PQ and RS are two transverse common tangent, then length of TCT Description: Description: 5289.png




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