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Translations Quiz 2

mathematics

 Which one of the following graphs represents $ABC$  after it translates twice to  $\begin{bmatrix}   2  \\   1  \\ \end{bmatrix}$ .

 Add the number of units to $x$ and $y$ coordinates of the given point, based on the given translation condition.

  $\text{Twice of } \begin{bmatrix}   2  \\   1  \\ \end{bmatrix} \\ \text{Which means moving }4\text{ units in positive }x\text{ direction } \\ \text{and }2\text{ units in positive y direction.} \\ \text{From the graph we can see } \text{ point A is at }(1, 5). \\ \text{ So, after translating }(4, 2)\text{ distance, we get} \\ A’ = (5, 7) \\ \text{Similarly, }B = (4, 1) \Rightarrow B’ = (8, 3) \\ C = (5, 3) \Rightarrow  C’ = (9, 5)$

 What is the coordinate of $A'$ if it is translated four times to $ \begin{bmatrix}   -2  \\   -2  \\ \end{bmatrix} \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/8d9eae46-f477-433e-bc98-7c2c6d72b6624504103430025677179.png"/>

  $\text{Four times of }\begin{bmatrix}   -2  \\   -2  \\ \end{bmatrix} \\ \text{Which means moving }8\text{ units in negative }x\text{ direction and }8\text{ units in negative y direction.} \\ \text{From the graph we can see } \text{that point A is at }(-3, 3).\text{ So, after translating }(-8, -8)\text{ distance, we get} \\ A’ = (-11, -5)$

 What will be the distance between the points $A$ and  $A' \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/4aaecf13-2348-4167-8441-d571b496432c4477322107774243388.png"/>

 Find the coordinates of $A$ and $A’$, then apply the distance formula.

  $\text{From the figure the coordinates of }A\text{ is }(2, 7)\text{ and the coordinates of }A’\text{ is }(-3, 8) \\ \text{Using the distance formula, } \\ d=\sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}}} \\ \text{So, the distance between }A\text{ and }A’\text{ is,} \\ d=\sqrt{{{\left( -3-2 \right)}^{2}}+{{\left( 8-7 \right)}^{2}}}=\sqrt{26}$

 What will be the position of  $A',B',C'$ after translation of $9$ left and $4$ upwards to the points $A\left( 2,5 \right),B\left( 4,3 \right),C\left( 5,3 \right)$.

  $\text{We have to move 9 units in negative } x \text{ direction } \\ \text{And 4 units in positive } y \text{ direction.} \\ \text{So, the }A\left( 2,5 \right),B\left( 4,3 \right),C\left( 5,3 \right)\text{ after translating, we get} \\ A'\left( -7,9 \right),B'\left( -5,7 \right),C'\left( -4,7 \right)$.

 What is the coordinate of  $A'$ if the translation is given as five times the coefficient of  $x$ and $y$   from the equation of line $AC$ in the image given below. $ \\ $<img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/ec5851d8-08d7-4314-8711-17ab6d77b9cf7411688394328630730.png"/>

 First find the equation of line $AC$ and find the coefficients of $x$ and $y$. Then multiply with 5 and then use this number to translate coordinates of $A$.

  $\text{Equation of line AC is, }(9, 5)\,\, (2, -1) \\ \left( y-{{y}_{1}} \right)=\dfrac{{{x}_{2}}-{{x}_{1}}}{{{y}_{2}}-{{y}_{1}}}\left( x-{{x}_{1}} \right) \\ \Rightarrow \left( y-5 \right)=\dfrac{2-9}{-1-5}\left( x-9 \right) \\ \Rightarrow -6y+30=-7x+63 \\ \Rightarrow 7x-6y=33 \\ \text{Five times the coefficient of } x \text{ and } y \text{ will be }(35, -30) \\ \text{So, the coordinates of } A’ \text{ after translation is,} A = (9, 5) \Rightarrow A’ = (44, -25)$.