Question

question 6 of 10

which of the following constructions were never accomplished by the greeks with only a straightedge and compass?

check all that apply.

a. tripling a square
b. doubling a cube
c. trisecting any angle
d. trisecting a line segment

Explanation:

Identify the constructibility of each option

Using the Compass and Straightedge Construction and Impossible Problems knowledge points, we evaluate each of the given geometric constructions to determine if they were impossible for the ancient Greeks using only an unmarked straightedge and a compass.

• A. Tripling a square: Tripling a square of area \(A\) means constructing a square of area \(3A\). The side length of the new square would be \(\sqrt{3}\) times the original side length. Since \(\sqrt{3}\) is a constructible number (obtainable via a right triangle with legs of length 1 and \(\sqrt{2}\)), this construction is possible.
• B. Doubling a cube: Using the Doubling the Cube knowledge point, doubling a cube of volume \(V\) requires constructing a cube of volume \(2V\). This requires constructing a side length of \(\sqrt[3]{2}\) times the original side length. Since \(\sqrt[3]{2}\) is an algebraic number of degree 3 over the rationals, it is not constructible using only a straightedge and compass. Thus, this was never accomplished.
• C. Trisecting any angle: Trisecting an arbitrary angle (such as \(60^\circ\) to get \(20^\circ\)) requires solving a cubic equation, which corresponds to a field extension of degree 3. Since compass and straightedge constructions can only generate field extensions of degree \(2^n\), trisecting an arbitrary angle is impossible. Thus, this was never accomplished.
• D. Trisecting a line segment: Trisecting any given line segment is easily accomplished using a standard parallel line construction (intercept theorem). Thus, this is possible.

Select the impossible constructions

The constructions that were never accomplished because they are mathematically impossible with only an unmarked straightedge and compass are:

• Doubling a cube (Option B)
• Trisecting any angle (Option C)

Answer:

• A. Tripling a square
• B. Doubling a cube (Correct answer)
• C. Trisecting any angle (Correct answer)
• D. Trisecting a line segment