Question

hexagon abcdef is inscribed in a circle with center h

1. what is the image of segment ab after a 120° clockwise rotation around h?
2. what is the image of segment ab after a reflection over line fc?

lines x and y are parallel.

1. what do you know about a,g,c, and e?
2. what do you know about h,b,f and d?
3. what do you know about a and d?
4. name a sequence of transformations that takes abc to abc

Explanation:

Step1: Analyze 120 - degree rotation of hexagon

A regular hexagon has rotational symmetry of order 6. A 120 - degree clock - wise rotation around the center \(H\) moves each vertex to the vertex two positions away in the clock - wise direction. Segment \(AB\) will map to segment \(DE\).

Step2: Analyze reflection of segment \(AB\) over line \(FC\)

The line \(FC\) is a line of symmetry for the hexagon. Segment \(AB\) and segment \(EF\) are symmetric with respect to the line \(FC\). So the image of segment \(AB\) after reflection over line \(FC\) is segment \(EF\).

Step3: Analyze angles formed by parallel lines \(x\) and \(y\)

For parallel lines \(x\) and \(y\) cut by a transversal:

• \(a\) and \(e\) are corresponding angles, so \(a = e\). \(a\) and \(g\) are vertical angles, so \(a = g\). \(c\) and \(e\) are alternate interior angles, so \(c = e\). Thus \(a = g=c = e\).
• \(h\) and \(d\) are corresponding angles, \(h\) and \(b\) are vertical angles, \(b\) and \(f\) are alternate interior angles. So \(h = b=f = d\).
• \(a\) and \(d\) are same - side interior angles, so \(a + d=180^{\circ}\).

Step4: Analyze transformation from \(\triangle ABC\) to \(\triangle A'B'C'\)

One possible sequence of transformations: First, translate \(\triangle ABC\) so that point \(C\) coincides with point \(C'\). Then, rotate the translated triangle around point \(C'\) to align the sides and vertices of the two triangles.

Answer:

1. Segment \(DE\)
2. Segment \(EF\)
3. \(a = g=c = e\)
4. \(h = b=f = d\)
5. \(a + d = 180^{\circ}\)
6. Translation followed by rotation.