Question

1. the figure shown, a square, was constructed using rigid motions, starting with the line segment with endpoints (2,4) and (6,8).

the line was rotated 90° clockwise about the point (2,4). the resulting image was then reflected across ____________ to complete the figure.

Explanation:

Step1: Recall the properties of rigid motions

Rigid motions (rotation, reflection) preserve the shape and size of the figure.

Step2: Analyze the rotation

When a line segment with endpoints \((2,4)\) and \((6,8)\) is rotated \(90^{\circ}\) clockwise about the point \((2,4)\), we can use the rotation formula. The vector from \((2,4)\) to \((6,8)\) is \(\langle6 - 2,8 - 4
angle=\langle4,4
angle\). After a \(90^{\circ}\) clockwise rotation about the origin, the vector \(\langle x,y
angle\) becomes \(\langle y,-x
angle\). But since we rotate about the point \((2,4)\), the new endpoint (after rotation) can be found.

Step3: Analyze the reflection

Let's consider the symmetry of the square. If we rotate the line segment \(90^{\circ}\) clockwise about \((2,4)\), and then we need to use reflection.
The mid - point of the original line segment and its rotated image (before reflection) can help us find the line of reflection.
The original line segment has endpoints \(A(2,4)\) and \(B(6,8)\). After a \(90^{\circ}\) clockwise rotation about \(A(2,4)\), let's find the new point.
The vector \(\overrightarrow{AB}=\langle4,4
angle\). After a \(90^{\circ}\) clockwise rotation about the origin, the vector becomes \(\langle4,- 4
angle\). So the new point (after rotation about \((2,4)\)) is \((2 + 4,4-4)=(6,0)\)
If we consider the symmetry of the square, the line of reflection is \(y = 4\).
Let's check:
The distance of a point \((x,y)\) from the line \(y = c\) is \(|y - c|\).
Take a point from the rotated line segment (before reflection) and its corresponding point in the square.
For example, if we have a point \((6,0)\) (after rotation), its reflection across \(y = 4\) is \((6,8)\) (using the formula \(y=2\times4 - 0=8\) for reflection across the line \(y = 4\): if we reflect a point \((x,y)\) across the line \(y = k\), the image is \((x,2k - y)\))

Answer:

the line \(y = 4\)