Congruent Transformation
Congruent transformation changes in position but no change in the size of an object is called congruent transformation .It is also called isometric transformation .Here, Reflection and Rotation are the examples of congruent transformation.
ReflectIon
A transformation in which a geometric figure is reflected across a line , creating a mirror image . That line is called the axis of a reflection .
Let's see the object given below and try jot down conclusion from the figure .
FIGURE
Conclusion
A rule which shifts an object is the reflection to the objects of same form each being at an equal distance from a fixed line .And this fixed line is known as mirror or axis of reflection .
Reflection using coordinates
As we know, the coordinate system has been well used .In coordinate plane, we can reflect the objects. Here, the coordinates of geometrical figures formed after the reflection about x-axis and y-axis . X-axis and Y-axis are known as the axis of reflection .
x-axis as the axis of reflection
Look at the following figure, and learn to find the coordinates of the image of a point in different quadrants when x-axis is the axis of reflection .
From the above figure, when a geometrical figure is reflected about x-axis as the axis of reflection, the x-coordinate of the image remains the same and the sign of y-coordinate of the image is changed .
FIGURE
y-axis as the axis of reflection: Look the following figure and learn the coordinate of the image of a point in different quadrants when y-axis is the axis of reflection .FIGURE
Look the following figure and learn to find the coordinates of the image of a point in different quadrants when y-axis is the axis of reflection .
FIGURE
Rotation
A transformation in which a plane figure turns around a fixed center point . In other words , one point on the plane the center of rotation , is fixed and everything else on the plane rotates about that point by a given angle .
Study the following figure and find out the idea of rotation of a point.
FIGURE
In the same way, each vertex of a geometrical figure are rotated through a certain angle in a certain direction about a given center of rotation , this figure is rotated through the same angle in the same direction about the same centre of rotation .To rotate this geometrical figure, these three conditions are required ;
A figure can be rotated in two directions.
Here, a figure rotates in opposite hand of a clock in an anti-clockwise direction.
Here, In the clockwise direction, we rotate a figure in the same direction of the rotation of the hand of a clock.
Now learn how to draw the image of a figure when it is rotated through the given angle in the given direction about a given centre of rotation.
Rotation through 90^{0} in anti-clockwise direction FIGURE
1.The dotted line of each vertex of figure should join to the centre of rotation
2.Draw 90^{0 }at o in each dotted lines with the help of a protractor in an anti-clockwise direction .
3.Cut off 0A’ = 0A, 0B’ = 0B and 0C’ = 0C.
4.Now, join A’, B’, and C’ .
Rotation through 90^{0} in clockwise direction
FIGURE
Rotation of geometrical figure using coordinates
Look the following figure and learn to find out the coordinates of the image of a point when it is rotated through 90^{0 }and 180^{0 }in anti-clockwise and clockwise directions about the centre of rotation at the origin .
Rotation through 90^{0} in anti-clockwise about origin
FIGURE
Rotation through 90^{0 }in clockwise about origin
FIGURE
In clockwise, a point is rotated through 90 in a direction about the origin as the centre of rotation, x and y-coordinates are exchanged by making the sign of y-coordinate just opposite.
Rotation through 180^{0 }in anti- clockwise and clockwise about origin
Here a point rotated through 180^{0} about the origin , then the coordinates of the image are the same in both directions. Learn the following figure .
FIGURE
In anti-clockwise, a point is rotated through 90^{0} or in a clockwise direction about the origin as the centre of rotation ,x and y-coordinates the image remains the same just by changing their signs.
Solution
Here, A(2,4) translated by (\(\frac{-5}{2}\)) and then ( 2-5, 4 +2) = (-3,6) , i.e C is obtained. The distance from B to D, N to M and C are parallel as well as equal. So, M maps N and D maps B.
Solution:
Join each of the vertex to the centre of rotation.
Draw angle of 90^{O} at O with OC, OA and OB in an anticlockwise direction.
Cut OA = OA' , OC' = OC' and OB = OB'
Join A' B' and C'
ΔA'B'C' is image of ΔABC
Solution
P(x,y) → P' (-y, x)
A(-3, 4) → A(4, 3)
B(4,3) →B(-4, 3)
Solution:
According to the question,
When ΔABC is rotated through 90º in an anti-clockwise direction about the origin.
A(3,2)→ A' (-3, 2)
B' (2,1)→B' (-2, 1)
C(2,3) →C' (-2, 3)
So, the coordinates of the vertices of image ΔA'B'C are A (-3, 2), B(-2, 1) and C(-2, 3).
The x-axis and y-axis in the co-ordinate system are called the ____________ .
When a point P( x,y) is reflected in x-axis it changes to P' _______________ .
When the point P(x, y) is reflected in y-axis it changes to P' ________________ .
The Congruent Transformation is also known as ________________ .
_______________ and ___________________ are the examples of congruent transformation.
In the reflection, the fixed line is called __________________ .
The object distance and image distance are always __________________ .
In anti-clockwise direction, we rotate a figure in ___________________________ .
While making the angle of 90^{o}, you should draw it in_______________ direction.
When the point is rotated through 180^{o} about origin, the coordinates of the image are _________________ in both direction.
ASK ANY QUESTION ON Transformation
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