Have you ever watched the hands of a clock move, seen a merry-go-round spin, or noticed the Earth turning? These all involve rotation—the spinning of an object around a fixed point or axis. Rotation is everywhere!
When you ride a merry-go-round or ride a bicycle, the wheels rotate, making everything move smoothly. The Earth itself rotates, which gives us day and night. Even when you stir your drink, you’re creating a small rotation!
Rotation is a transformation that moves an object around a fixed point but its size and shape are not changed. Rotations are sometimes called turns. The point around which a rotation occurs is called the centre of rotation, and the distance a shape turns is called the angle of rotation
Using a ruler, draw a straight line from point \(A\) to \(O\) and also from point \(A'\) to \(O\) as shown below. Measure the distance \(OA\) and \(OA'\) and record your results. What do you notice?
Similarly, draw a straight line from point \(B\) to \(O\) and also from point \(B'\) to \(O\text{.}\) Measure the distance \(OB\) and \(OB'\) and record your results. What do you notice?
Finally, draw a straight line from point \(C\) to \(O\) and also from point \(C'\) to \(O\text{.}\) Measure the distance \(OC\) and \(OC'\) and record your results. What do you notice?
The coordinates of the vertices for triangle \(PQR\) that can be graphed in the coordinate plane are \((-8,-6)\text{,}\)\((-2,-6)\) and \((-5,-3)\) as shown below. The triangle is rotated through \(90^\circ\) in a clockwise direction about the origin to produce triangle \(P'Q'R'\text{.}\)Copy the figure and draw triangle \(P'Q'R'\)
Similary join point \(O\) to \(B'\) and \(B\) and \(O\) to \(A'\) and \(A\)and measusure \(\angle\, BOB'\) and \(\angle\,AOA'\text{.}\) What do you notice?
Triangle \(A'B'C'\) is the image of triangle \(ABC\) after a rotation.The centre and angle of rotation can be found by drawing the perpendicular bisectors of the lines between two sets of points, \(C\) and \(C'\) and \(B\) and \(B'\) or \(A\) and \(A'\)
The point where two perpendicular bisectors intersect is called \(\textbf{the centre of rotation.}\) To find \(\textbf{ the angle of rotation}\text{,}\) join \(C'\) and \(C\) to the centre of rotation and measure the angle between these lines.
Also join point \(Y\) to \(Y'\) and construct a perpendicular bisector to \(YY'\) as shown below. Mark the point of intersection of perpendicular bisectors \(O\)
Rotation on different planes" refers to the concept of rotating an object or point around various axes within different planes in a three-dimensional space.
The figure below shows triangle \(ABC\) and its images after rotations about the origin with different angles of rotation (\(90^\circ\text{,}\)\(180^\circ\text{,}\)\(270^\circ\) and \(360^\circ\)).
Consider a point \(A\, (4,3)\) .We are required to finding the coordinates of its image after a Rotation taking the centre to be \((1,2)\) and angle of rotation to be \(90^\circ\) ;
Considering the same point \(P\text{,}\) but now the angle of rotation to be \(180^\circ\text{;}\) To find the coordinates of its image we follow the following steps:
A triangle \(ABC\) with coordinates \(A\,(2,1)\text{,}\)\(B\, (3,2)\) and \(C\, (3,4)\) is rotated through the centre and angle of \(90^\circ\) in a clockwise direction.Find the coordinates of its image
Since triangle \(ABC\) is rotated in a clockwise direction with an angle of \(90^\circ\) through the origin, then the angle of rotation is \(-90^\circ\)
According to the rule, If we have our points \((p,q)\) which will be mapped to \((q,-p)\) if rotated through the centre and angle of rotation is \(-90^\circ\)
Describe the rotation which maps the rectangle whose verticies are \(P\, (2,2)\text{,}\)\(Q\, (6,2)\text{,}\)\(R\,(6,4)\) and \(S\, (2,4)\) onto a rectangle whose verticies are \(P'\,(2,-2)\text{,}\)\(Q'\,(2,-6)\text{,}\)\(R'\,(4,-6)\) and \(S'\, (4,-2)\)
Find the coordinates of the verticies of the image of a triangle whose verticies are \(P\, (-4,6)\text{,}\)\(Q\, (-4,2)\) and \(R\, (-2,2)\) when rotated about the origin through:
The parallelogram whose verticies are \(A\, (4,4)\text{,}\)\(B\, (8,4)\text{,}\)\(C\, (2,2)\) and \(D\, (6,2)\) is rotated to give an image whose verticies are \(A'\, (4,-2)\text{,}\)\(B'\, (4,-6)\text{,}\)\(C'\, (2,0)\) and \(D'\, (2,-4)\text{.}\) Find the centre and angle of rotation.
The number of times the tracing of the star fits onto the printed copy in one complete turn is \(5 \) times. This is called the order of rotational symmetry, that is, the number of times the figure fits onto itself in one complete turn \((360^\circ).\)
The straight line around which the object is rotated is called axis of rotation. In the activity, the strings represents the axes of rotational symmetry for the cuboid.
Find the other axes of rotation and order of rotational symmetry of the regular tetrehedron given one of the axes from \(A\) and passing at the center of the face \(BDC.\text{.}\)
Place a protractor at the line \(AD\) with the centre of the protractor at \(D\) and measure \(90^\circ.\) Using a ruler draw \(DA'\) such that \(AD\,=\,DA'.\)
Rotation is a type of transformation that repositions an object but preserves the shape and size of the object. Thus, rotation produces congruent figures.
