NCERT Class 10 Mathematics Formula CBSE Board Sample Problems Part 6 (For CBSE, ICSE, IAS, NET, NRA 2022)

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Triangles

Types of Triangles
S. no.TermDescriptions
1CongruenceTwo Geometric figure are said to be congruence if they are exactly same size and shape

Symbol used is

Two angles are congruent if they are equal

Two circle are congruent if they have equal radii

Two squares are congruent if the sides are equal

2Triangle CongruenceTwo triangles are congruent if three sides and three angles of one triangle is congruent to the corresponding sides and angles of the other
Triangle Congruence

Corresponding sides are equal

Corresponding angles are equal

We write this as

The above six equalities are between the corresponding parts of the two congruent triangles. In short form this is called C. P. C. T

We should keep the letters in correct order on both sides

3Inequalities in Triangles(1) In a triangle angle opposite to longer side is larger

(2) In a triangle side opposite to larger angle is larger

(3) The sum of any two sides of the triangle is greater than the third side

In triangle ABC

Different Criterion for Congruence of the Triangles

Different Criterion for Congruence of the Triangles
NCriterionDescriptionFigures and expression
1Side angle Side (SAS) congruenceTwo triangles are congruent if the two sides and included angles of one triangle is equal to the two sides and included angle

It is an axiom as it cannot be proved so it is an accepted truth

ASS and SSA type two triangles may not be congruent always

Two Triangles
If following condition

Then

2Angle side angle (ASA) congruenceTwo triangles are congruent if the two angles and included side of one triangle is equal to the corresponding angles and side

It is a theorem and can be proved

Two Triangles
If following condition

Then

3Angle-angle side (AAS) congruenceTwo triangles are congruent if the any two pair of angles and any side of one triangle is equal to the corresponding angles and side

It is a theorem and can be proved

Two Triangles
If following condition

Then

4Side-Side-Side (SSS) congruenceTwo triangles are congruent if the three sides of one triangle is equal to the three sides of the another
Two Triangles
If following condition

Then

5Right angle hypotenuse side (RHS) congruenceTwo right triangles are congruent if the hypotenuse and a side of the one triangle are equal to corresponding hypotenuse and side of the anotherIf following

condition

Then

Some Important Points on Triangles

Some Important Points on Triangles
TermsDescription
OrthocenterPoint of intersection of the three altitude of the triangle
EquilateralTriangle whose all sides are equal and all angles are equal to 600
MedianA line Segment joining the corner of the triangle to the midpoint of the opposite side of the triangle
AltitudeA line Segment from the corner of the triangle and perpendicular to the opposite side of the triangle
IsoscelesA triangle whose two sides are equal
CentroidPoint of intersection of the three median of the triangle is called the centroid of the triangle
In centerAll the angle bisector of the triangle passes through same point
CircumcenterThe perpendicular bisector of the sides of the triangles passes through same point
Scalene triangleTriangle having no equal angles and no equal sides
Right TriangleRight triangle has one angle equal to 900
Obtuse TriangleOne angle is obtuse angle while other two are acute angles
Acute TriangleAll the angles are acute

Similarity of Triangles

Similarity of Triangles
S. noPoints
1Two figures having the same shape but not necessarily the same size are called similar figures.
2All the congruent figures are similar but the converse is not true.
3If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points. then the other two sides are divided in the same ratio
4If a line divides any two sides of a triangle in the same ratio. Then the line is parallel to the third side.

Different Criterion for Similarity of the Triangles

Different Criterion for Similarity of the Triangles
NCriterionDescriptionExpression
1Angle-Angle angle (AAA)Two triangles are similar if corresponding angle are equal similarityIf following

condition

Then

Then

2Angle-angleTwo triangles are similar if the two If following (AA) similarity corresponding angles are equal as condition by angle property third angle will be also equalIf following

condition

Then

Then

Then

3Side-side-side (SSS) SimilarityTwo triangles are similar if the sides of one triangle is proportional to the sides of other triangleIf following

Condition

Then

Then

4Side-Angle- Side (SAS) similarityTwo triangles are similar if the one angle of a triangle is equal to one angle of other triangles and sides including that angle is proportionalIf following Condition

And

Then

Area of Similar Triangles

If the two triangle ABC and DEF are similar

Then

Pythagoras Theorem

Pythagoras Theorem
S. noPoints
1If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse. Then the triangles on both sides of the perpendicular are similar to the whole triangle and to each other.
2In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (Pythagoras Theorem) .

3If in a triangle, square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle

Arithmetic Progression

Arithmetic Progression
S. noTermsDescription
1Arithmetic ProgressionAn arithmetic progression is a sequence of numbers such that the difference of any two successive members is a constant

Examples

(1) 1, 5, 9,13, 17 …

(2) 1, 2,3, 4,5 …

2Common difference of the APthe difference between any successive members is a constant and it is called the common difference of AP

(1) If are the terms in AP then

(2) We can represent the general form of AP in the form

Where a is first term and d is the common difference

3 term of Arithmetic Progression term
4Sum of nth item in Arithmetic

Progression

Or

Trigonometry

Trigonometry
S. noTermsDescription
1What is TrigonometryTrigonometry from Greek trigonon, ( “triangle,” and metron, “measure” ) is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged during the 3rd century BC from applications of geometry to astronomical studies.

Trigonometry is most simply associated with planar right-angle triangles (each of which is a two-dimensional triangle with one angle equal to 90 degrees) . The applicability to non-right-angle triangles exists, but since any non-right-angle triangle (on a flat plane) can be bisected to create two right-angle triangles, most problems can be reduced to calculations on right-angle triangles. Thus, the majority of applications relate to right-angle triangles

2Trigonometric Ratio՚sIn a right-angle triangle ABC where
Trigonometric Ratio՚s

We can define following term for angle A

Base: Side adjacent to angle

Perpendicular: Side Opposite of angle

Hypotenuse: Side opposite to right angle

We can define the trigonometric ratios for

angle A as

Notice that each ratio in the right-hand column is the inverse, or the reciprocal, of the ratio in the left-hand column.

3Reciprocal of functionsThe reciprocal of sin A is cosec A; and vice versa.

The reciprocal of cos A is sec A

And the reciprocal of tan A is cot A

These are valid for acute angles.

We can define tan

And Cot A

4Value of sin and cosIs always less 1
5Trigonometric ration from another angleWe can define the trigonometric ratios for angle C as
Trigonometric Ration from Another Angle

6Trigonometric ratios of complimentary angles

7Trigonometric identities

Trigonometric Ratios of Common Angles

We can find the values of trigonometric ratio՚s various angle

We Can Find the Values of Trigonometric Ratio՚s Various Angle
Angles (A)Cos ATan ACosec ASec ACot A
010Not defined1Not defined
2
11
2
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