The rule $T_{5,-0.5^{\circ}} R_{0,180^{\circ}}(x, r)$ is applied to $\Delta F G H$ to produce $\Delta F ^{\prime} G ^{\prime \prime} H ^{\prime \prime}$.
What are the coordinates of vertex $F ^{\prime \prime}$ of $\Delta F ^{\prime \prime} G ^{\prime \prime} H ^{\prime \prime}$ ?
(4,-1.5)
(4,-0.5)
(-1.5,4)
( -0.5,4 )
Answer (2)
Apply the rotation R 0 , 18 0 ∘ to point F ( 4 , − 1.5 ) , resulting in F ′ ( − 4 , 1.5 ) .
Apply the translation T 5 , − 0. 5 ∘ to point F ′ ( − 4 , 1.5 ) , resulting in F ′′ ( − 4 + 5 , 1.5 − 0.5 ) .
Calculate the final coordinates of F ′′ as ( 1 , 1 ) .
The final coordinates of vertex F ′′ are ( 1 , 1 ) .
Explanation
Understand the Transformation We are given a point F ( 4 , − 1.5 ) and a transformation T 5 , − 0. 5 ∘ R 0 , 18 0 ∘ ( x , y ) . This means we first rotate the point ( x , y ) by 18 0 ∘ about the origin, and then translate the resulting point by ( 5 , − 0.5 ) .
Apply the Rotation The rotation R 0 , 18 0 ∘ ( x , y ) maps ( x , y ) to ( − x , − y ) . Applying this to F ( 4 , − 1.5 ) , we get F ′ ( − 4 , 1.5 ) .
Apply the Translation The translation T 5 , − 0. 5 ∘ ( x , y ) maps ( x , y ) to ( x + 5 , y − 0.5 ) . Applying this to F ′ ( − 4 , 1.5 ) , we get F ′′ ( − 4 + 5 , 1.5 − 0.5 ) = ( 1 , 1 ) .
Determine the Final Coordinates Therefore, the coordinates of vertex F ′′ are ( 1 , 1 ) . Comparing this to the given options, we see that none of the options match our result. However, let's re-examine the problem statement. The rotation is R 0 , 18 0 ∘ and the translation is T 5 , − 0. 5 ∘ . We apply the rotation first, then the translation. So, F ( 4 , − 1.5 ) becomes F ′ ( − 4 , 1.5 ) after rotation. Then, F ′ ( − 4 , 1.5 ) becomes F ′′ ( − 4 + 5 , 1.5 − 0.5 ) = ( 1 , 1 ) after translation. The correct answer should be ( 1 , 1 ) . Since this is not among the options, there might be a typo in the question or the options.
Examples
Understanding transformations like rotations and translations is crucial in computer graphics for manipulating objects in 2D or 3D space. For example, in video games, when a character moves or rotates, these transformations are applied to update the character's position and orientation on the screen. Similarly, in image editing software, you can rotate, scale, or translate images using these transformations.
The coordinates of vertex F ′′ after the transformations are ( 1 , 1 ) . However, this answer does not match any of the provided multiple-choice options, indicating a potential mistake in those options. Therefore, based on the calculations, none of the options are correct.
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