Problem 1: Cutting a Square-Based Cake into Two Congruent Parts

A square-based cake can be divided into two congruent parts in a variety of ways, all depending on the type of symmetry involved.

Symmetry-based Approach

A square exhibits several symmetry properties:

1. Vertical symmetry: Cutting along the vertical axis through the center of the square.
2. Horizontal symmetry: Cutting along the horizontal axis through the center of the square.
3. Diagonal symmetry (two cuts): Cutting along either of the two diagonals of the square.

Thus, there are four basic ways to cut the cake into two congruent parts:

• 1 vertical cut
• 1 horizontal cut
• 2 diagonal cuts

Conclusion: There are 4 ways to cut a square-based cake into two congruent parts.

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Problem 2: Using Eight Straight Lines to Form a Square and Four Congruent Equilateral Triangles

To solve this, imagine the geometric layout needed to create the required shapes using straight lines:

1. Drawing the Configuration:

   • Draw a large square using four of the straight lines (each side of the square will consist of one line).
   • Inside this square, position the remaining four lines as diagonals to divide the interior of the square into four smaller congruent equilateral triangles.

   Since the problem states that the eight lines must form both a square and four congruent equilateral triangles, the diagonals must be arranged in such a way that they intersect at angles of 60° within the square.

2. Proving It Works:

   • Each side of the square is composed of one straight line, and since the diagonals are straight lines, they must intersect to form the equilateral triangles.
   • By symmetry and the properties of equilateral triangles, it is possible to arrange the diagonals in a way that each triangle has equal sides formed by segments of the lines.
   • For a mathematical proof, you would need to calculate the angle between the intersecting diagonals and verify that it corresponds to the angles of equilateral triangles.

Conclusion: By carefully positioning the diagonals within the square, the eight straight lines can be arranged to form both a square and four equilateral triangles.

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Follow-up Questions:

1. Can you describe how rotational symmetry affects the number of ways to cut the cake?
2. What if the cake were a cube instead of a square-based cake? How would the number of cuts change?
3. How would the solution to the second problem change if the lines were not required to be of equal length?
4. What types of congruences would apply if the cake were not square, but rectangular?
5. How do you prove that each triangle in the second problem is equilateral using basic geometry?

Tip: When solving geometry problems involving symmetry, look for operations (rotations, reflections) that simplify your reasoning.