Question

rotation
spinning a shape around a certain point

Answer:

The problem here is about the geometric transformation of rotation (spinning a shape around a point). To analyze or perform a rotation, we can use the following general steps for a rotation about a point (let's assume the point is the origin for simplicity, but it can be any point \((h,k)\)):

Step 1: Recall the rotation formula

For a rotation of \(\theta\) degrees about the origin \((0,0)\), the transformation of a point \((x,y)\) is given by:

• For a counterclockwise rotation:

\(x' = x\cos\theta - y\sin\theta\)
\(y' = x\sin\theta + y\cos\theta\)

• For a clockwise rotation, we can use \(\theta = -\theta\) in the above formulas, so:

\(x' = x\cos\theta + y\sin\theta\)
\(y' = -x\sin\theta + y\cos\theta\)

If the rotation is about a point \((h,k)\) (not the origin), we first translate the point so that \((h,k)\) becomes the origin: \((x - h, y - k)\), perform the rotation, and then translate back: \((x' + h, y' + k)\).

Step 2: Identify the shape and the center of rotation

From the image, we have a shape (a triangle - like figure) and a center of rotation (the '+' sign). Let's assume we want to find the coordinates of the rotated shape. First, we need to determine the coordinates of the vertices of the original shape relative to the center of rotation.

Step 3: Determine the angle of rotation

Looking at the arc (the curved arrow), we can estimate the angle of rotation. If it's a 90 - degree counterclockwise rotation (a common case in such diagrams), \(\theta=90^\circ\) or \(\frac{\pi}{2}\) radians. \(\cos(90^\circ)=0\), \(\sin(90^\circ) = 1\)

Step 4: Apply the rotation formula

Let's say we have a vertex of the original shape at \((x,y)\) relative to the center of rotation \((h,k)\). After translation, the point is \((x - h,y - k)\). For a 90 - degree counterclockwise rotation:
\(x'_{translated}=(x - h)\cos(90^\circ)-(y - k)\sin(90^\circ)=(x - h)\times0-(y - k)\times1=-(y - k)\)
\(y'_{translated}=(x - h)\sin(90^\circ)+(y - k)\cos(90^\circ)=(x - h)\times1+(y - k)\times0=(x - h)\)

Then we translate back:
\(x'=x'_{translated}+h=-(y - k)+h=h - y + k\)
\(y'=y'_{translated}+k=(x - h)+k=x - h + k\)

If we want to describe the rotation in general, the key steps are: identify the center, the angle, and then apply the rotation transformation (either using the formula for rotation about the origin and then adjusting for the center, or using the formula for rotation about a general point).

If we were to find the image of a specific point, say the original point is \((x_1,y_1)\) and the center of rotation is \((h,k)\) with an angle of \(\theta\), we would follow the above steps.

For example, if the center of rotation is \((0,0)\) (the '+' sign is at the origin) and we have a point \((2,1)\) on the original shape and we rotate it 90 degrees counterclockwise:

Step 1: Identify \(\theta = 90^\circ\), \(x = 2\), \(y=1\)

Step 2: Apply the counterclockwise rotation formula

\(x'=2\cos(90^\circ)-1\sin(90^\circ)=2\times0 - 1\times1=- 1\)
\(y'=2\sin(90^\circ)+1\cos(90^\circ)=2\times1+1\times0 = 2\)

So the image of the point \((2,1)\) after a 90 - degree counterclockwise rotation about the origin is \((-1,2)\)

If the question was to describe the rotation or find the image of the shape, the above steps would be used.

If we assume the problem is to find the image of the shape after rotation (say 90 degrees counterclockwise about the center), the general process is:

1. Find the coordinates of each vertex of the original shape relative to the center of rotation.
2. Apply the rotation formula (depending on the direction and angle of rotation) to each vertex.
3. Plot the new vertices to get the rotated shape.

For example, if the original shape has vertices \(A(x_1,y_1)\), \(B(x_2,y_2)\), \(C(x_3,y_3)\) and the center of rotation is \(O(h,k)\):

• Translate each vertex: \(A'(x_1 - h,y_1 - k)\), \(B'(x_2 - h,y_2 - k)\), \(C'(x_3 - h,y_3 - k)\)
• Rotate each translated vertex: For a 90 - degree counterclockwise rotation, \(A''(-(y_1 - k),x_1 - h)\), \(B''(-(y_2 - k),x_2 - h)\), \(C''(-(y_3 - k),x_3 - h)\)
• Translate back: \(A'''(-(y_1 - k)+h,(x_1 - h)+k)\), \(B'''(-(y_2 - k)+h,(x_2 - h)+k)\), \(C'''(-(y_3 - k)+h,(x_3 - h)+k)\)

These new points \(A'''\), \(B'''\), \(C'''\) form the rotated shape.

If the angle of rotation is different, say 180 degrees, \(\cos(180^\circ)=- 1\), \(\sin(180^\circ)=0\)

For a 180 - degree rotation about the origin, the transformation is \(x'=-x\), \(y'=-y\)

And for a 180 - degree rotation about \((h,k)\), we have:

• Translate: \((x - h,y - k)\)
• Rotate 180 degrees: \((-(x - h),-(y - k))=(-x + h,-y + k)\)
• Translate back: \((-x + h+h,-y + k + k)=(-x + 2h,-y + 2k)\)

If the problem is to define rotation, the answer is: Rotation is a geometric transformation that involves spinning a two - dimensional shape around a fixed point (called the center of rotation) by a certain angle (in degrees or radians). The shape retains its size and shape, but its position and orientation change according to the angle and direction (clockwise or counterclockwise) of the rotation.