Triangle $Q R S$ has vertices $Q(8,-6), R(10,5)$, and $S(-3,3)$. What are the coordinates of the vertices of the image of the triangle after a translation of $T_{-7.6,4.3}(x, y)$?
$Q^{\prime}=$
$R^{\prime}=$
$S^{\prime}=$
Answer (2)
After applying the translation T − 7.6 , 4.3 to triangle QRS with vertices Q ( 8 , − 6 ) , R ( 10 , 5 ) , and S ( − 3 , 3 ) , the new coordinates are Q ′ ( 0.4 , − 1.7 ) , R ′ ( 2.4 , 9.3 ) , and S ′ ( − 10.6 , 7.3 ) . The translations involve subtracting 7.6 from the x-coordinates and adding 4.3 to the y-coordinates of each vertex. Therefore, the final vertices of the image of the triangle are these calculated points.
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Apply the translation T − 7.6 , 4.3 ( x , y ) to each vertex of triangle QRS .
Calculate the new coordinates for vertex Q ′ : Q ′ = ( 8 − 7.6 , − 6 + 4.3 ) = ( 0.4 , − 1.7 ) .
Calculate the new coordinates for vertex R ′ : R ′ = ( 10 − 7.6 , 5 + 4.3 ) = ( 2.4 , 9.3 ) .
Calculate the new coordinates for vertex S ′ : S ′ = ( − 3 − 7.6 , 3 + 4.3 ) = ( − 10.6 , 7.3 ) .
The coordinates of the translated vertices are Q ′ ( 0.4 , − 1.7 ) , R ′ ( 2.4 , 9.3 ) , S ′ ( − 10.6 , 7.3 ) .
Explanation
Problem Analysis We are given triangle QRS with vertices Q ( 8 , − 6 ) , R ( 10 , 5 ) , and S ( − 3 , 3 ) . We need to find the coordinates of the vertices of the image of the triangle after a translation of T − 7.6 , 4.3 ( x , y ) . This means that each point ( x , y ) will be translated to ( x − 7.6 , y + 4.3 ) .
Finding Q' To find the coordinates of Q ′ , we apply the translation to Q ( 8 , − 6 ) :
Q ′ = ( 8 − 7.6 , − 6 + 4.3 ) = ( 0.4 , − 1.7 ) So, Q ′ = ( 0.4 , − 1.7 ) .
Finding R' To find the coordinates of R ′ , we apply the translation to R ( 10 , 5 ) :
R ′ = ( 10 − 7.6 , 5 + 4.3 ) = ( 2.4 , 9.3 ) So, R ′ = ( 2.4 , 9.3 ) .
Finding S' To find the coordinates of S ′ , we apply the translation to S ( − 3 , 3 ) :
S ′ = ( − 3 − 7.6 , 3 + 4.3 ) = ( − 10.6 , 7.3 ) So, S ′ = ( − 10.6 , 7.3 ) .
Final Answer The coordinates of the vertices of the image of the triangle after the translation are: Q ′ = ( 0.4 , − 1.7 ) R ′ = ( 2.4 , 9.3 ) S ′ = ( − 10.6 , 7.3 )
Examples
Translations are used in computer graphics to move objects around the screen. For example, if you are designing a video game, you might use translations to move a character from one location to another. Each point of the character is translated by the same vector, so the character moves as a whole without changing its shape or orientation. This is also used in robotics, where robots need to move objects from one place to another. The robot's arm can be programmed to translate an object by a certain vector, ensuring the object is moved accurately and efficiently.
