Question
If \( \mathrm{P}=(5,4) \), find the image of P under the following rotation. \( 90^{\circ} \) counterclockwise about the origin \[ ([?],[]) \] Enter the number that belongs in the green box.
Ask by Floyd Macdonald. in the United States
Mar 09,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The image of point P after a \(90^\circ\) counterclockwise rotation is \((-4, 5)\). The number that belongs in the green box is \(-4\).
Solution
Given the transformation for a \(90^\circ\) counterclockwise rotation about the origin, any point \((x, y)\) maps to
\[
(-y, x)
\]
For the point \(\mathrm{P}=(5,4)\), we have:
\[
x = 5 \quad \text{and} \quad y = 4.
\]
Thus, the new coordinates are:
\[
x' = -4 \quad \text{and} \quad y' = 5.
\]
So, the image of \(\mathrm{P}\) under the rotation is:
\[
(-4, 5)
\]
The number that belongs in the green box (the first coordinate in the transformed point) is \(-4\).
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Simplify this solution The Deep Dive
To find the image of point \( \mathrm{P}=(5,4) \) under a \( 90^{\circ} \) counterclockwise rotation about the origin, we can use the rotation formula for a point \( (x, y) \). The new coordinates after the rotation can be calculated using the transformation: \[ (x', y') = (-y, x) \] Applying this to point \( P \), \[ x' = -y = -4 \] \[ y' = x = 5 \] Thus, the image of point \( P \) after a \( 90^{\circ} \) counterclockwise rotation is \( (-4, 5) \). Therefore, the number that belongs in the green box is -4. Now we have \( [ -4, 5 ] \).
