Question

which geometric construction is shown below?

a. doubling the square
b. tripling the cube
c. tripling the square
d. doubling the cube

Explanation:

Analyze the geometric diagram

The diagram displays a smaller square on the left. A diagonal is drawn across this smaller square. Using this diagonal as a side length, a larger square is constructed, tilted to the right.

Apply geometric principles

Let the side length of the smaller square be \(s\). The area of this smaller square is:

$$A_1 = s^2$$

The diagonal of the smaller square, which forms the side length of the larger square, is calculated using the Pythagorean theorem:

$$d = s\sqrt{2}$$

The area of the larger square constructed on this diagonal is:

$$A_2 = d^2 = (s\sqrt{2})^2 = 2s^2$$

Relate to historical concepts

Comparing the two areas shows that the larger square has exactly twice the area of the smaller square:

$$A_2 = 2A_1$$

This classic geometric construction is known as Doubling the square, famously discussed in Plato's dialogue Meno to demonstrate how to construct a square with twice the area of a given square.

Match with options

• Option A: Doubling the square (Matches our finding)
• Option B: Tripling the cube
• Option C: Tripling the square
• Option D: Doubling the cube

Answer:

• A. Doubling the square (Correct answer)
• B. Tripling the cube
• C. Tripling the square
• D. Doubling the cube