Question
Parallelogram RSTU is rotated 45° clockwise using the origin as the center of rotation.
On a coordinate plane, parallelogram R S T U has points (2, 3), (5, 3), (7, 1), (4, 1).
Which graph shows the image of RSTU?
On a coordinate plane, parallelogram R prime S prime T prime U prime has points (negative 1, 4), (1, 6), (4, 6), (2, 4).
On a coordinate plane, parallelogram R prime S prime T prime U prime has points (3.5, 1), (5.5, negative 1), (5.5, negative 4), (3.5, negative 2).
On a coordinate plane, parallelogram R prime S prime T prime U prime has points (2, negative 3), (5, negative 3), (7, negative 1), (4, negative 1).
On a coordinate plane, parallelogram R prime S prime T prime U prime has points (3.2, 4.8), (5.5, 2.5), (5.5, negative 0.5), (3.2, 1.8).
On a coordinate plane, parallelogram R S T U has points (2, 3), (5, 3), (7, 1), (4, 1).
Which graph shows the image of RSTU?
On a coordinate plane, parallelogram R prime S prime T prime U prime has points (negative 1, 4), (1, 6), (4, 6), (2, 4).
On a coordinate plane, parallelogram R prime S prime T prime U prime has points (3.5, 1), (5.5, negative 1), (5.5, negative 4), (3.5, negative 2).
On a coordinate plane, parallelogram R prime S prime T prime U prime has points (2, negative 3), (5, negative 3), (7, negative 1), (4, negative 1).
On a coordinate plane, parallelogram R prime S prime T prime U prime has points (3.2, 4.8), (5.5, 2.5), (5.5, negative 0.5), (3.2, 1.8).
Answers
Answered by
GPT-4o mini
To find the image of parallelogram RSTU after a 45° clockwise rotation about the origin, we can apply the rotation transformation to each of the vertices of the parallelogram.
The points of the parallelogram RSTU are:
- R(2, 3)
- S(5, 3)
- T(7, 1)
- U(4, 1)
The formula for rotating a point (x, y) by an angle θ (in radians) around the origin is:
- x' = x * cos(θ) + y * sin(θ)
- y' = -x * sin(θ) + y * cos(θ)
For a 45° rotation, θ = 45° = π/4 radians. The cosine and sine of 45° are both √2/2.
Now we can apply the rotation to each point:
1. **Point R(2, 3)**:
- x' = 2 * (√2/2) + 3 * (√2/2) = (2 + 3) * (√2/2) = 5(√2/2) ≈ 3.54
- y' = -2 * (√2/2) + 3 * (√2/2) = (3 - 2)(√2/2) = (1)(√2/2) = √2/2 ≈ 0.71
So, R' ≈ (3.54, 0.71).
2. **Point S(5, 3)**:
- x' = 5 * (√2/2) + 3 * (√2/2) = 8 * (√2/2) ≈ 5.66
- y' = -5 * (√2/2) + 3 * (√2/2) = -2 * (√2/2) = -√2 ≈ -1.41
So, S' ≈ (5.66, -1.41).
3. **Point T(7, 1)**:
- x' = 7 * (√2/2) + 1 * (√2/2) = 8 * (√2/2) ≈ 5.66
- y' = -7 * (√2/2) + 1 * (√2/2) = -6 * (√2/2) = -3√2 ≈ -4.24
So, T' ≈ (5.66, -4.24).
4. **Point U(4, 1)**:
- x' = 4 * (√2/2) + 1 * (√2/2) = 5 * (√2/2) ≈ 3.54
- y' = -4 * (√2/2) + 1 * (√2/2) = -3 * (√2/2) ≈ -2.12
So, U' ≈ (3.54, -2.12).
Putting this all together, the approximate coordinates of the rotated parallelogram R'S'T'U' will be:
- R' ≈ (3.54, 0.71)
- S' ≈ (5.66, -1.41)
- T' ≈ (5.66, -4.24)
- U' ≈ (3.54, -2.12)
Now we can evaluate the provided options to check which correlates to the calculated results.
The correct answer is that none of the provided points exactly match the calculations because they all are approximate due to the calculations involved in rotation.
But, if we analyze further, if we focus more closely on approximate values, one of the given choices may resemble values after a transformation hinting at some potential translations or element manipulations depending on context, but rotation with exact calculations is essential.
In conclusion, for practical purposes, choose the one most approximate to the rotation output based on previously calculated transformations or hints that indicate other corresponding points from the provided multiple-choice options, confirming the roughly comprehended manipulations through observational graphics/alignments from each graph's values within user visualization of RSTU to R'S'T'U'.
The points of the parallelogram RSTU are:
- R(2, 3)
- S(5, 3)
- T(7, 1)
- U(4, 1)
The formula for rotating a point (x, y) by an angle θ (in radians) around the origin is:
- x' = x * cos(θ) + y * sin(θ)
- y' = -x * sin(θ) + y * cos(θ)
For a 45° rotation, θ = 45° = π/4 radians. The cosine and sine of 45° are both √2/2.
Now we can apply the rotation to each point:
1. **Point R(2, 3)**:
- x' = 2 * (√2/2) + 3 * (√2/2) = (2 + 3) * (√2/2) = 5(√2/2) ≈ 3.54
- y' = -2 * (√2/2) + 3 * (√2/2) = (3 - 2)(√2/2) = (1)(√2/2) = √2/2 ≈ 0.71
So, R' ≈ (3.54, 0.71).
2. **Point S(5, 3)**:
- x' = 5 * (√2/2) + 3 * (√2/2) = 8 * (√2/2) ≈ 5.66
- y' = -5 * (√2/2) + 3 * (√2/2) = -2 * (√2/2) = -√2 ≈ -1.41
So, S' ≈ (5.66, -1.41).
3. **Point T(7, 1)**:
- x' = 7 * (√2/2) + 1 * (√2/2) = 8 * (√2/2) ≈ 5.66
- y' = -7 * (√2/2) + 1 * (√2/2) = -6 * (√2/2) = -3√2 ≈ -4.24
So, T' ≈ (5.66, -4.24).
4. **Point U(4, 1)**:
- x' = 4 * (√2/2) + 1 * (√2/2) = 5 * (√2/2) ≈ 3.54
- y' = -4 * (√2/2) + 1 * (√2/2) = -3 * (√2/2) ≈ -2.12
So, U' ≈ (3.54, -2.12).
Putting this all together, the approximate coordinates of the rotated parallelogram R'S'T'U' will be:
- R' ≈ (3.54, 0.71)
- S' ≈ (5.66, -1.41)
- T' ≈ (5.66, -4.24)
- U' ≈ (3.54, -2.12)
Now we can evaluate the provided options to check which correlates to the calculated results.
The correct answer is that none of the provided points exactly match the calculations because they all are approximate due to the calculations involved in rotation.
But, if we analyze further, if we focus more closely on approximate values, one of the given choices may resemble values after a transformation hinting at some potential translations or element manipulations depending on context, but rotation with exact calculations is essential.
In conclusion, for practical purposes, choose the one most approximate to the rotation output based on previously calculated transformations or hints that indicate other corresponding points from the provided multiple-choice options, confirming the roughly comprehended manipulations through observational graphics/alignments from each graph's values within user visualization of RSTU to R'S'T'U'.