How to find slope of a line?

Find the slope of the line where the points are ( 9,5) and (7,3) by using straight line tools.

Solution:

The formula for slope m = (y₁-y₂) / (x₁-x₂).

M = (5-3) / (9 -7).

M = 2 / 2.

= 1.

Thus the slope of line is 1.

General rules for multiplication

General rules for multiplication:

Positive x positive = positive

Negative x negative = positive

Positive x negative =negative

Negative x positive = negative

Positive x positive = positive:

Example:

x 4 =12

This two number are positive integers. So, answer is positive.

Negative x negative = positive

For example,

-4 x -3 = 12

This two number are negative integers .so, answer is positive.

Positive x negative =negative

For example,

4 x 3 =-12

One positive integer and one negative integer .so, answer is negative.

Negative x positive = negative

For example,

-4 x -3 =-12

One negative integer and one positive integer.so, answer is negative.

Solve the algebraic expression

Solve the algebraic expression:

4(x -1) + 2y - 5(x -y -4) + 5

Multiplying the integer terms

= 4x - 4 + 2y -5x + 5y + 20 + 5

Grouping the above terms

= -x + 7y + 21

Geometry Problem and Solution

Problem: Can two vertically opposite angles be supplement to each other? Draw a diagram to illustrate your answer


Let AB and CD bisect each other at right angles as shown in figure.







( Vertically opposite angles can be
supplements to each other)

Arithmetic Progressions

Arithmetic Progressions:
Consider the following lists of numbers :
(i) 1, 2, 3, 4, . . .
(ii) 100, 70, 40, 10, . . .
(iii) – 3, –2, –1, 0, . . .
(iv) 3, 3, 3, 3, . . .
(v) –1.0, –1.5, –2.0, –2.5, . . .
Each of the numbers in the list is called a term. In (i), each term is 1 more than the term preceding it. In (ii), each term is 30 less than the term preceding it. In (iii), each term is obtained by adding 1 to the term preceding it. In (iv), all the terms in the list are 3 , i.e., each term is obtained by adding (or subtracting) 0 to the term preceding it. In (v), each term is obtained by adding – 0.5 to (i.e., subtracting 0.5 from) the term preceding it. In all the lists above, we see that successive terms are obtained by adding a fixed number to the preceding terms. Such list of numbers is said to form an Arithmetic Progression ( AP ). So, an arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term.

Slope of a Line

Slope of a Line:
A line in a coordinate plane forms two angles with the x-axis, which are supplementary.
The angle (say) θ made by the line l with positive direction of x-axis and measured anti clockwise
is called the inclination of the line. Obviously 0° ≤ θ ≤ 180° (below Fig ). We observe that lines parallel to x-axis, or coinciding with x-axis, have inclination of 0°. The inclination of a vertical line (parallel to coinciding with y-axis) is 90°.


Definition 1 If θ is the inclination of a line l, then tan θ is called the slope or gradient of the line l. The slope of a line whose inclination is 90° is not defined. The slope of a line is denoted by m. Thus, m = tan θ, θ ≠ 90° It may be observed that the slope of x-axis is zero and slope of y-axis is not defined.

Perimeter and Area of a Circle

Perimeter and Area of a Circle:
circumference of a circle bears a constant ratio with its diameter. This constant ratio
is denoted by the Greek letter π (read as ‘pi’). In other words

Circumference/Diameter=π
circumference = π × diameter = π × 2r (where r is the radius of the circle) = 2πr


The great Indian mathematician Aryabhatta (A.D. 476 – 550) gave an approximate
value of π. He stated that π =62832/20000 which is nearly equal to 3.1416.
However, for practical purposes, we generally take the value of π as 22/7 or 3.14, approximately

area of the circle =1/2× 2πr × r = πr2