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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
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
- XOX’ and YOY’ are respectively known as the x-axis and y-axis.
- O(0, 0) is called origin.
- The ordered pair of real numbers (x, y) is called Cartesian coordinates of point P.
- The x – co-ordinate is called abscissa which is the distance of the point from the y-axis parallel to the x-axis.
- The y – coordinate is called ordinate which is the distance of the point from the x-axis parallel to the y-axis.
- x – coordinate of a point on the y-axis is zero.
- 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.
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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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
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Now we are to discuss the explanations of the point shortcut tricks of above questions
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
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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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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