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This is one of our many quizzes on Rotations. Click on this to access more such competitive quizzes on Geometry.

Rotations Quiz 2

mathematics

 A triangle is made by $3$ points. The points are A$(5,6)$, B $(4,3)$ and C $(7,8)$. If a new figure is made by rotating these points at an angle of  ${270}^{\circ}$ clockwise. Find the coordinates of the new triangle.

 If the original coordinates are given by $(x,y)$ the coordinates after rotating ${270}^{\circ}$ clockwise will be $(-y,x)$.

  If the original coordinates are given by $(x,y)$ the coordinates after rotating ${270}^{\circ}$ clockwise will be $(-y,x)$ $ \\ $ Here for the point $(5,6)$, $x=5$  $ \\ $ And $ \\  y=6 \\  \Rightarrow -y=-6 \\ $ So the new coordinate will be $(-6,5)$  $ \\ $ Similarly other points will be $(4,3)\equiv (-3,4)$ and $(7,8)\equiv (-8,7)$ 

 A Rod on the side of a machine component is given by a straight line represented by the equation $y=x+1$ is rotated ${90}^{\circ}$ clockwise. What is the new equation of the rod?

 In a complex plane, any curve is the locus of $z=x(t)+iy(t)$ where $x(t)$ and $y(t)$ are real functions of parameter $t$.

  Considering a parameter $t$ $x=t$ and $y=t+1$  $ \\ $ In a complex plane, any curve is the locus of $z=x(t)+iy(y)$ where $x(t)$ and $y(t)$ are real function of parameter $t$ $ \\ $ Here $z=t+i(t+1)$ and t ranges over all real numbers $ \\ $ Rotating by ${90}^{\circ}$ is the multiplication by  $ \\ {{e}^{i{{90}^{\circ}}}}={{e}^{i\dfrac{\pi }{2}}}=i \\ $  The rotated line is $ \\ z=i(t+i(t+1))=-(t+1)+it \\ $  So $ \\  x=-(t+1) \\  y=t \\  \Rightarrow y=-x+1$

 A piece of puzzle in the Tetris game is represented as $\left( 1,-1 \right)$  is rotated around ninety degrees by the player I in clockwise direction, what will be the new way to represent it? 

 If the original coordinates are given by $(x,y)$ the coordinates after rotating ninety degrees clockwise will be $(y,-x)$.

If the original coordinates are given by $(x,y)$ the coordinates after rotating ninety degrees clockwise will be $(y,-x)$ $ \\ $ Here, $ \\  x=1 \\  \Rightarrow -x=-1 \\ $ And $y=-1$ $ \\ $ So the new coordinates will be $(-1,-1)$ 

 The line of a figure is denoted by the equation $3x+2y=1$. It is rotated about a point $(-1,2)$ by a right angle. What will be the equation of this new line?

 Rotation of a line by ${90}^{\circ}$ gives you a perpendicular line, and if the point of rotation is on the initial line, then your task is simply finding a perpendicular line through that point.

 Rotation of a line by ${90}^{\circ}$ gives you a perpendicular line, and if the point of rotation is on the initial line, then your task is simply finding a perpendicular line through that point. $ \\  3(-1)+2(2)=1 \\  \Rightarrow (-1,2)\text{ lies on }3x+2y=1 \\ $ Solving for $y$ will get your initial slope: $ \\  3x+2y=1 \\  \Rightarrow 2y=-3x+1 \\  \text{slope}=-\dfrac{3}{2} \\ $ The perpendicular slope is the negative reciprocal: $ \\ m\bot =\dfrac{-1}{m}=\dfrac{2}{3} \\ $  We now have the rotated/perpendicular line as: $\\ y=\dfrac{2}{3}x+b$  Going through $(-1,2) \\  \Rightarrow (2)=\dfrac{2}{3}(-1)+b \\  \Rightarrow b=\dfrac{8}{3} \\  \Rightarrow y=\dfrac{2}{3}x+\dfrac{8}{3} \\ $ Rearranging the equation as $3y=2x+8$ 

 A civil engineer finds that a lane represented by a line with coordinates  $(4,8)$ and $(9,10)$ had been rotated ninety degrees clockwise earlier. What were the original coordinates?

 If the original coordinates are given by $(x,y)$ the coordinates after rotating ninety degrees clockwise will be $(y,-x)$ $ \\ $ Here for the point$(4,8)$, $  \\   x=4 \\  \Rightarrow -x=-4 \\ $ And $y=-8$ $ \\ $ So the new coordinates will be $(8,-4)$  $ \\ $ And for the point $(9,10)$, $ \\  x=9 \\  \Rightarrow -x=-9 \\ $ And $y=10$ $ \\ $ So the new coordinates will be $(8,-4)$ and $(10,-9)$ 