If the translation T maps point $A (-3,1)$ onto point $A ^{\prime}(5,5)$, what is the translation T ?
A. $T_{<8,4>}$
B. $T_{<8,6>}$
C. $T_{\langle 2,4\rangle}$
D. $T_{\langle 2,6\rangle}$
Answer (2)
The translation T maps point A ( − 3 , 1 ) onto point A ′ ( 5 , 5 ) .
We find the translation vector T = such that A + T = A ′ .
Solving the equations − 3 + x = 5 and 1 + y = 5 , we get x = 8 and y = 4 .
The translation T is }}"> T < 8 , 4 > .
Explanation
Problem Analysis Let's analyze the problem. We are given a point A ( − 3 , 1 ) and its image A ′ ( 5 , 5 ) after a translation T . We need to find the translation vector T .
Setting up equations Let the translation vector be T = . The translation maps point A to point A ′ , so we have: A + T = A ′ ( − 3 , 1 ) + = ( 5 , 5 ) This gives us two equations: − 3 + x = 5 1 + y = 5
Solving for x and y Now, let's solve for x and y :
For x :
x = 5 + 3 x = 8 For y :
y = 5 − 1 y = 4
Finding the translation vector Therefore, the translation vector T is "> < 8 , 4 > .
Final Answer The translation T is }"> T < 8 , 4 > . So the answer is A.
Examples
Translations are used in computer graphics to move objects around the screen. For example, if you have a character at position (-3, 1) and you want to move it to (5, 5), you would apply the translation <8, 4> to its coordinates. This ensures the character moves smoothly and accurately to the desired location.
The translation vector that maps point A(-3, 1) to A'(5, 5) is T = ⟨8, 4⟩. Therefore, the correct answer is A. T_{⟨8, 4⟩}.
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