10 CBSE Co-ordinate Geometry Ex. 7.1 Q. 3

10 CBSE Co-ordinate Geometry Ex. 7.1 sum ( question) no. 3. After getting the distances using distance formula. The sum is about how to decide whether the points are collinear. Finding square roots of 5, 212 & 265 by long division will take lot of time, so the alternative method which is the best short cut. 

You may even find by geometrical construction 9 CBSE by taking suitable scale (by dividing by a suitable perfect square say 25). However this method also is time-consuming and needs lot of practice to master it.

How to know if a given number is a perfect square or not?

For a perfect square following numbers should be last 2 right-most digits:

0	1	4	9	16	25
	21	24	29	36
	41	44	49	56
	61	64	69	76
	81	84	89	96

write squares of numbers

you can keep on writing squares of natural numbers from 1. Write 1 and to its right sq. of 1. Go to next line write next number i.e. 2 below first '1' written. Now add to 1 written in square column number written in natural number column. Add the latest number written in natural number column to the previous sum. Continue like this till you want.
algebra behind:
( a + 1) sq = a sq + 2a +1 = a sq + a + ( a + 1 )

Trigonometry

Trigonometry:
Old Hindustani Singer Adores His Collection Of Albums, Tunes.
The first letter of every word can be used to remember the 3 basic trigonometric ratios of sine, cosine & tangent.
Opposite side / Hypotenuse = Sin
Adjacent side / Hypotenuse = Cos
Opposite side / Adjacent side = Tan

Cotangent is complementary tangent. So reciprocal of Cot is Tan.
Reciprocal of Secant is Cosine.
Reciprocal of Cosec is Sine.
C & C do not go together.
so also S & S.

In right angled triangle,
(1) Side opposite to 90 deg. is longest and it is called Hypotenuse. In short hyp.
In triangle ABC, if /_ B = 90 deg, then
AB sq + BC sq = AC sq
(2) Note letter B gets repeated. angle B is right angle.
(3) On R.H.S. B disappears and remaining letters get connected.

Hemant - Gajanan Cevian Theorem

Angle Divider Theorem©®
Refer attachments

New method of dividing a number by 7, when a number is divisible by 7

Short cut method of dividing by 7 when a number is divisible by 7.
This method is based on divisibility test for 7.
When a number passes this test , use the calculation done to find the quotient.
1) Calculate quotient for last step.
2) For a step, add the subtracted number and the right-most digit for that step.
3) Write this addition below quotient in first step such that only one digit comes extra at the right.
4) Repeat the steps 2) and 3) till you reach the dividend (the number you wanted to divide by 7).

http://mathandmultimedia.com/2012/02/29/divisibility-by-7-and-its-proof/

Application of Proability

There are 2 adjacent bus stops. A person A wants to reach a particular place by travelling by bus. For the destination, buses of 3 routes stop at 1st stop and bus of 1 route stops at other stop. Distance between 2 bus stops is 120 feet, At what distance should A stand from 1st bus stop.

Solution:
Total no. of bus routes for the particular destination = 4.
Buses of no. of bus routes stooping at 1st stop = 3
Therefore, Probability that A will get bus from 1st stop is 3/4.
{Had there been no bus stopping at 2nd bus stop for his destination he would have stopped at 1st bus stop only.}
He should stand nearer to 1st bus stop. So distance from 1st stop will be = (1-3/4)*120=  (1/4)*120 =30 feet from 1st bus stop.