The rule $R_{0,90^{\circ}} T_{-1,1}(x, y)$ is applied to $\Delta BCD$ to produce $\Delta B ^{\prime \prime} C ^{\prime \prime} D ^{\prime \prime}$. Point B " of the final image is at $(-4, 1)$.
What are the coordinates of point $B$ on the pre-image?
$(-2,-5)$
$(0,-5)$
$(2,3)$
$(5,0)$
Answer (2)
Apply the translation T − 1 , 1 ( x , y ) to point B, resulting in point B' with coordinates ( x − 1 , y + 1 ) .
Apply the rotation R 0 , 9 0 ∘ to point B', resulting in point B'' with coordinates ( − ( y + 1 ) , x − 1 ) .
Solve the system of equations − ( y + 1 ) = − 4 and x − 1 = 1 to find x and y .
The coordinates of point B are ( 2 , 3 ) .
Explanation
Applying the transformations Let the coordinates of point B be ( x , y ) . The transformation T − 1 , 1 ( x , y ) translates point B to B' with coordinates ( x − 1 , y + 1 ) . The rotation R 0 , 9 0 ∘ rotates point B' to B'' with coordinates ( − ( y + 1 ) , x − 1 ) . We are given that B'' is at ( − 4 , 1 ) . Therefore, we have the equations: − ( y + 1 ) = − 4 and x − 1 = 1 .
Solving for x and y Solving the system of equations: From the equation x − 1 = 1 , we get x = 1 + 1 = 2 .
From the equation − ( y + 1 ) = − 4 , we get y + 1 = 4 , so y = 4 − 1 = 3 .
Finding the coordinates of B Therefore, the coordinates of point B are ( 2 , 3 ) .
Examples
Understanding transformations like translations and rotations is crucial in computer graphics for creating animations and special effects. For example, when designing a video game, developers use these transformations to move characters and objects around the screen, rotate them, and create realistic movements. By applying a series of transformations, they can achieve complex animations and interactions within the game world, making the experience more engaging and visually appealing for the player.
The coordinates of point B on the pre-image before transformations are ( 2 , 3 ) . After applying the translation and rotation, we confirmed that the resulting point B ′′ is at ( − 4 , 1 ) . Therefore, the correct answer is ( 2 , 3 ) .
;
