Triangle DEF has coordinates D(-3,-2), E(-1,-2), F(-2,-4). The triangle is translated so its image has coordinates D'(1,-4), E'(3,-4), F(2, -6). Which statement shows the rule for the translation?
A. (x, y)→(x-4, y-2)
B. (x, y)→(x-4, y+2)
C. (x, y)→(x+4, y+2)
D. (x, y)→(x+4, y-2)
Answer (2)
Calculate the horizontal translation by subtracting the x-coordinate of the original point from the x-coordinate of the translated point.
Calculate the vertical translation by subtracting the y-coordinate of the original point from the y-coordinate of the translated point.
Express the translation rule as ( x , y ) → ( x + a , y + b ) , where 'a' is the horizontal translation and 'b' is the vertical translation.
The translation rule is ( x , y ) → ( x + 4 , y − 2 ) .
Explanation
Analyze the problem We are given triangle DEF with vertices D(-3,-2), E(-1,-2), F(-2,-4) and its image D'(1,-4), E'(3,-4), F'(2, -6) after a translation. Our goal is to find the translation rule (x, y) -> (x+a, y+b) that maps DEF to D'E'F'.
Calculate horizontal translation To find the horizontal translation 'a', we calculate the difference between the x-coordinates of D' and D: a = D ′ . x − D . x = 1 − ( − 3 ) = 1 + 3 = 4 .
Calculate vertical translation To find the vertical translation 'b', we calculate the difference between the y-coordinates of D' and D: b = D ′ . y − D . y = − 4 − ( − 2 ) = − 4 + 2 = − 2 .
State the translation rule Therefore, the translation rule is (x, y) -> (x+4, y-2).
Verify the rule Let's verify the rule by applying it to point E(-1, -2): E' = (E.x + 4, E.y - 2) = (-1 + 4, -2 - 2) = (3, -4), which matches the given coordinates of E'. Now, let's verify the rule by applying it to point F(-2, -4): F' = (F.x + 4, F.y - 2) = (-2 + 4, -4 - 2) = (2, -6), which matches the given coordinates of F'.
Final Answer The translation rule that maps triangle DEF to triangle D'E'F' is (x, y) -> (x+4, y-2).
Examples
Imagine you're designing a video game where a character needs to move from one location to another on the screen. Applying a translation is like telling the character to move a certain number of steps to the right (horizontal translation) and a certain number of steps up or down (vertical translation). For example, if the character's initial position is (x, y) and you want to move it 5 steps to the right and 3 steps down, the new position would be (x+5, y-3). This concept is fundamental in game development for character movement, object placement, and creating dynamic environments.
The translation rule for triangle DEF translating to its new coordinates is ( x , y ) → ( x + 4 , y − 2 ) .
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