Here is the design for the flag of Trinidad and Tobago.

Describe a sequence of translations, rotations, and reflections that take the lower left triangle to the upper right triangle.

Here is a picture of an older version of the flag of Great Britain. There is a rigid transformation that takes Triangle 1 to Triangle 2, another that takes Triangle 1 to Triangle 3, and another that takes Triangle 1 to Triangle 4.

<p>An image of an older version of the flag of Great Britain. The flag is a rectangle with a vertical length about twice the width. Red stripes divide the flag in half vertically and horizontally. White stripes connect the vertices along diagonals, crossing behind the red stripes. The remaining area is composed of 8 blue right triangles. At the top of the flag, 2 large right triangles line up on either side of the vertical red stripe by their shorter square sides, so that they are mirror images of each other. At the bottom of the flag, 2 large right triangles line up on either side of the vertical red stripe by their shorter square sides, so that they are mirror images of each other. At the left side, 2 small right triangles line up on either side of the horizontal red stripe by their longer square sides so that they are mirror images of each other. The triangle above the red stripe is labeled 1; the triangle below the red strip is labeled 3. At the right side, 2 small right triangles line up on either side of the horizontal red stripe by their longer square sides so that they are mirror images of each other. The triangle above the red stripe is labeled 2; the triangle below the red strip is labeled 4.</p>

1. Measure the lengths of the sides in Triangles 1 and 2. What do you notice?
2. What are the side lengths of Triangle 3? Explain how you know.
3. Do all 8 triangles in the flag have the same area? Explain how you know.

1. Which of the lines in the picture is parallel to line \(\ell\)? Explain how you know.

   

   Three lines, m, k and l, cut by a transversal, p.  Lines k and l will not intersect no matter how far they extend. Line m appears to be angled towards lines k and l on the right so that it would intersect them at a point not shown.  

2. Explain how to translate, rotate or reflect line \(\ell\) to obtain line \(k\).
3. Explain how to translate, rotate or reflect line \(\ell\) to obtain line \(p\).

Point \(A\) has coordinates \((3,4)\). After a translation 4 units left, a reflection across the \(x\)-axis, and a translation 2 units down, what are the coordinates of the image?

Here is triangle \(XYZ\):

Draw these three rotations of triangle \(XYZ\) together.

1. Rotate triangle \(XYZ\) \(90^\circ\) clockwise around \(Z\).
2. Rotate triangle \(XYZ\) \(180^\circ\)around \(Z\).
3. Rotate triangle \(XYZ\) \(270^\circ\) clockwise around \(Z\).