Triangle RST has vertices $R (2,0), S (4,0)$, and $T (1,-3)$. The image of triangle RST after a rotation has vertices $R ^{\prime}(0,-2), S ^{\prime}(0,-4)$, and $T ^{\prime}(-3,-1)$. Which rule describes the transformation?
$R_{0,90^{\circ}}$
$R_{0,180^{\circ}}$
$R_{0,270^{\circ}}$
$R_{0,360^{\circ}}$
Answer (2)
A 90-degree rotation maps (x,y) to (-y,x). Applying this to R(2,0) gives (0,2).
A 180-degree rotation maps (x,y) to (-x,-y). Applying this to R(2,0) gives (-2,0).
A 270-degree rotation maps (x,y) to (y,-x). Applying this to R(2,0) gives (0,-2).
The 270-degree rotation maps all three vertices correctly, so the answer is R 0 , 27 0 ∘ .
Explanation
Analyze the problem and data We are given triangle RST with vertices R ( 2 , 0 ) , S ( 4 , 0 ) , and T ( 1 , − 3 ) . The image of triangle RST after a rotation has vertices R ′ ( 0 , − 2 ) , S ′ ( 0 , − 4 ) , and T ′ ( − 3 , − 1 ) . We need to determine the rule that describes the transformation, choosing from R 0 , 9 0 ∘ , R 0 , 18 0 ∘ , R 0 , 27 0 ∘ , and R 0 , 36 0 ∘ .
Check each rotation for point R Let's analyze the transformation of point R(2,0) to R'(0,-2). We will check each of the possible rotations.
A 90-degree rotation maps (x,y) to (-y,x). Applying this to R(2,0) gives (0,2), which is not R'(0,-2).
A 180-degree rotation maps (x,y) to (-x,-y). Applying this to R(2,0) gives (-2,0), which is not R'(0,-2).
A 270-degree rotation maps (x,y) to (y,-x). Applying this to R(2,0) gives (0,-2), which is R'(0,-2).
A 360-degree rotation maps (x,y) to (x,y). Applying this to R(2,0) gives (2,0), which is not R'(0,-2).
Verify the 270-degree rotation for points S and T Since a 270-degree rotation maps R(2,0) to R'(0,-2), we need to check if it also maps S(4,0) to S'(0,-4) and T(1,-3) to T'(-3,-1).
Applying a 270-degree rotation to S(4,0) gives (0,-4), which is S'(0,-4).
Applying a 270-degree rotation to T(1,-3) gives (-3,-1), which is T'(-3,-1).
Conclusion Since the 270-degree rotation maps all three vertices correctly, the rule that describes the transformation is a 270-degree rotation about the origin.
Examples
Understanding rotations is crucial in many real-world applications, such as computer graphics, robotics, and physics. For example, when designing a robotic arm, engineers need to calculate the precise rotations required for the arm to reach a specific point in space. Similarly, in computer graphics, rotations are used to create realistic animations and 3D models. The principles of coordinate geometry and transformations are fundamental to these applications, allowing for accurate and efficient manipulation of objects in space.
The transformation rule that describes the rotation of triangle RST is a 270-degree rotation about the origin, written as R 0 , 27 0 ° ◯ . This is confirmed by checking how each vertex of the triangle transforms under this rotation. The vertices match the transformed points precisely.
;
