• ## Geometry Axioms & Theorems

Class 9  LINES AND ANGLES  Axiom 1  If a rays stands on a line , then the sum of two adjacent angles so formed is 180 0 Axiom 6.2 : If the sum of two adjacent angles is 180°, then the non-common arms of the angles form a line.  Theorem 6.1 : If two lines […]

• ## CBSE 10 – Mathematics

Educational Tools Study Tools Ask, Answer & Discuss Chapter Videos Watch In English & Hindi Speed Notes Synopsis For Quick Revision Full Notes Topicwise Detailed Notes with Key Terms, Mind Maps & More … E-Book Chapterwise Texbook Assignment Tools Testing Tools To Assign, assess & Analyse NCERT Solutions Solved Exercises NCERT Exemplar HOTS Questions & […]

• ## Board Class – Subject

Recap & Test Your Previous Understanding Watch In English & Hindi English Version Hindi Version Course Content Speed Notes Synopsis for revision E – Book Chapterwise Textbooks Assignments NCERT Solutions Solved Exercises NCERT Exemplar HOTS Questions & Answers Assessments Descriptive Tests & Worksheets Objective Quizzes Academic Forums Clarify Doubts Ask, Answer & Discuss Resources Hub […]

• ## Assess – Chapter Template

Educational Tools Study Assess Interact Quiz Objective Assessment Worksheet Creative Assessment Assessment Standard Assessment Question Bank Board Exam Questions Clarify Doubts ↗ Get a virtual tour of the museum. Ideal for schools and events. Resources Hubs ↗ Stay updated and see our current exhibitions here. Academic Clubs ↗ Get to know our opening times, ticket […]

• ## Study – Chapter Template

Educational Tools Study Assess Interact Pre – Requisites Basics Required 1. Chapter Video Watch in English & Hindi Speed Notes 🗒 Qick Revision These metals are moderately reactive. They exists as sulphides or carbonates in nature. Before reduction, metal sulphides and carbonates must be converted into metal oxides. Sulphide ores are These metals are moderately […]

Cartesian System

Coordinate Axes:

The position of a point in a plane is fixed by selecting the axes of reference which are formed by two number lines intersecting each other at right angles, so that their zeroes coincide.

The horizontal number line is called x-axis and vertical number line is called y axis.

A point that lies on X Axis is (x,0)

A point that lies on Y Axis is (0,y)

Equation of Y Axis is x = 0

Equation of X Axis is y = 0

Equation of a lne parallel to Y Axis is x = a

Equation of a lne parallel to X Axis is y = a

Equation of a lne perpendicular to X Axis is X = a

Equation of a lne perpendicular to X Axis is X = a

The point of intersection of the two lines is called origin.

is the x-axis and Y1OY is the y-axis. These coordinate axes are also called rectangular axes as they are perpendicular to each other.

Rectangular coordinates are ordered pairs in which the first element is called the abscissa and the second element is called the ordinate.

● In the first quadrant, x is + ve and y is + ve

● In the second quadrant, x is – ve and y is + ve

● In the third quadrant, x is – ve and y is – ve

● In the fourth quadrant, x is + ve and y is -ve.

Distance Formula:

Example:

Example:

Collinearity of three points:

Three points P, Q and R are said to be collinear, if they lie in the same straight line.

i.e., PR = PQ + QR

i.e., PQ = PR + RQ

i.e., QR = QP + PR

If three points are not collinear, they always form a triangle.

Special Polygons:

(i) In Case of Triangle

(a) a right-angled triangle, if sum of squares of any two sides is equal to square of third side.

(b) an equilateral triangle, if all the three sides are equal.

(c) an isosceles triangle, if any two sides are equal.

(a) parallelogram, if opposite sides are equal and diagonals are not equal.

(b) rectangle, if opposite sides are equal and diagonals are equal.

(c) square, if all the four sides are equal and diagonals are equal.

(d) rhombus, if all the four sides are equal and diagonals are not equal.

Section Formula (Internal division only)

Midpoint Formula:

Point Dividing Two points in K : 1 Ratio:

Note:

If k is positive, the point divides the given points internally.

If k is Negative, the point divides the given points externally

Coordinates of the centroid of a triangle:

Points of Trisection:

If a line segment is divided into three equal parts by two points, the points are said to be the points of trisection.

In the given figure, the points R and S divide the line segment PQ into three equal parts i.e., PR=RS=SQ. The points R and S are said to be points of trisection.

Area of a Triangle:

The area of the triangle formed by the points

is calculated by the following expression.

Area of ∆PQR =

Area of a quadrilateral can be found by splitting up the quadrilateral into two triangles and sum up their areas.

Thus, area of quadrilateral PQRS = area of ∆PQR+ area of ∆PRS

Condition for collinearity of three points :

Three given points will be collinear, if the area of the triangle formed by these points is zero.

Rule to prove that three given points are collinear:

Step 1. Find the area of the triangle formed by the given points.

Step 2. Show that the area of the triangle formed by the given points is zero.

* The coordinates of the origin are O(0,0)

* The coordinates of any point on x-axis are (x, 0)

i.e., y=0 or ordinate is zero.

* The coordinates of any point on y – axis are (0, y) i.e., x=0 or abscissa is zero.

error: Content is protected !!