If a translation maps point $A(-3,1)$ to point $A'(5,5)$, the translation is:
A. $(x+2, y+4)$
B. $(x+8, y+6)$
C. $(x+8, y+4)$
D. $(x+2, y+6)$
Answer (2)
The translation maps point A ( − 3 , 1 ) to A ′ ( 5 , 5 ) .
We find the translation vector by setting up equations for the x and y coordinates: − 3 + a = 5 and 1 + b = 5 .
Solving these equations gives a = 8 and b = 4 .
The translation is therefore ( x + 8 , y + 4 ) .
Explanation
Problem Analysis Let's analyze the problem. We are given a point A ( − 3 , 1 ) that is mapped to A ′ ( 5 , 5 ) by a translation. We need to find the translation vector. A translation shifts every point by the same amount in the x and y directions.
Setting up the equations Let the translation be ( x + a , y + b ) . This means that the x-coordinate of the original point is increased by a , and the y-coordinate is increased by b . So, we have:
( − 3 + a , 1 + b ) = ( 5 , 5 )
Forming equations Now we can set up two equations:
− 3 + a = 5 1 + b = 5
Solving for a Solving for a :
a = 5 + 3 a = 8
Solving for b Solving for b :
b = 5 − 1 b = 4
Finding the translation Therefore, the translation is ( x + 8 , y + 4 ) .
Final Answer The translation that maps point A ( − 3 , 1 ) to point A ( 5 , 5 ) is ( x + 8 , y + 4 ) .
Examples
Translations are used in computer graphics to move objects around the screen. For example, if you have a character in a game at position (-3, 1) and you want to move it to position (5, 5), you would apply the translation (x+8, y+4) to the character's coordinates. This ensures that the character moves smoothly and accurately to the desired location.
The translation that maps point A ( − 3 , 1 ) to A ′ ( 5 , 5 ) is determined by finding the change in x and y coordinates. We find that the translation is given by ( x + 8 , y + 4 ) . Thus, the correct answer is option C: ( x + 8 , y + 4 ) .
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