## The Point Shortcut tricks

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## Introduction

In this article, we are going to discuss The Point Shortcut Tricks which contains the following contents such as co-ordinates axes, distance formula, section formula, types of centres like centroid, incentre, circumcentre, and orthocentre, etc.

In this The Point Shortcut Tricks, we will discuss Basic concepts, shortcut tricks, MCQ – Test series, and Explanations with shortcut tricks

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## Basic Concepts

Today, we are to discuss the following basic concepts such as distance formula, section formula, midpoint formula, centres like centroid, incentre, circumcentre, and orthocentre, etc.

## Rectangular Cartesian Co-ordinates

Let XOX’ and YOY’ be two fixed straight lines, which intersect at right angles at ‘O’ (0, 0), then

1. XOX’ and YOY’ are respectively known as the x-axis and y-axis.
2. O(0, 0) is called origin.
3. The ordered pair of real numbers (x, y) is called Cartesian coordinates of point  P.
4. The x – co-ordinate is called abscissa which is the distance of the point from the y-axis parallel to the x-axis.
5. The y – coordinate is called ordinate which is the distance of the point from the x-axis parallel to the y-axis.
6. x – coordinate of a point on the y-axis is zero.
7. Y – co-ordinate of a point on the x-axis is zero.

The co-ordinate axes divide a plane into the four parts which are known as quadrants.

As the region of the areas  XOY, X’OY, X’OY’and XOY’ are called first, second, third, and fourth quadrants respectively.

## Polar Co-ordinates

The polar coordinates of the point are specified as (r, ϴ) Where r = distance of the point from the origin (radius vector) and  ϴ = angle between the radius vector and x-axis which is known as a Vectorial angle.

## Relationship between Cartesian and polar Co-ordinates

If (x, y) be the Cartesian coordinates of a point P lying in a plane and (r, ϴ) be its polar coordinates with respect to origin ‘O’ as the pole and x-axis as the initial line, then

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## Shortcut Tricks

In this article, we will also discuss the shortcut tricks related to all the above concepts.

Shortcut Trick To identify the right-angled triangle from the given three vertices of type A(a, b), B(a, c), and C(d, b) of an ABC

Here we observed that

x co-ordinates of vertex A = x co-ordinates of vertex B

y co-ordinates of vertex A = y co-ordinates of vertex C

In such a condition given triangle must be a right-angled triangle and right-angled at the vertex A(a, b)

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## MCQ – Test Series

Here we have the MCQ test series about The Point Shortcut Tricks look at below

Q.1 The mid points of sides of a triangle are (2, 4), (4, 6) and (6, 10). Then the co-ordinates of its vertices are

• (5, 8), (0, 2), (8, 12)
• (4, 8), (0, 0), (8, 12)
• (6, 8), (0, 0), (8, 10)
• None of these

Q.2 Number of integral interior points of the triangle having vertices (0, 0), (0, 31), and (31, 0) is

• 535
• 635
• 435
• 735

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## Explanations

Now we are to discuss the explanations of the point shortcut tricks of above questions

Solution.1

The midpoints of sides of a triangle are P(2, 4), Q(4, 6), and R(6, 10). Then the coordinates of its vertices are A(4, 8), B(0, 0), C(8, 12). See below shortcut trick

In a ABC ,if P, Q, R are the midpoints of sides of triangle, then its vertices A, B, C are given by

A = P + R – Q         B = P + Q – R      C = R + Q – P

solution.2

Number of integral interior points of the triangle having vertices (0, 0), (0,31) and (31, 0) is given by    = (31- 1)/2 x (31-2) = 15 x 29 = 435

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## conclusion

Finally, we say on behalf of the above discussion that The Point Shortcut Tricks article is very most important for us

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Thank You

### Nitin Sharma

Hello Dosto ....This is Nitin Sharma working as a Mathematician, Writer, You Tuber, Blogger, Lyricist, and Author. By this platform, I express my knowledge and my passion by writing Articles on Mathematical Concepts. I hope you are enjoying it. Thank You!