Question

question 25 of 25

which of these constructions is impossible using only a compass and straightedge?

a. doubling the square
b. trisecting a right angle
c. bisecting any angle
d. doubling the cube

Explanation:

Analyze option A

Using the Doubling the Square knowledge point
Doubling a square of side \(s\) requires constructing a square with area \(2s^2\), which means its side length must be \(s\sqrt{2}\). Since \(\sqrt{2}\) is a constructible number, this construction is possible.

Analyze option B

Using the Angle Trisection knowledge point
Trisecting a right angle (\(90^\circ\)) requires constructing an angle of \(30^\circ\). Since a \(30^\circ\) angle can be constructed (for example, by constructing an equilateral triangle and bisecting one of its \(60^\circ\) angles), this specific trisection is possible.

Analyze option C

Using the Angle Bisector Theorem knowledge point
Bisecting any given angle is a standard, universally possible classical construction using a compass and straightedge.

Analyze option D

Using the Doubling the Cube knowledge point
Doubling a cube of side \(s\) requires constructing a cube with volume \(2s^3\), which means its side length must be \(s\sqrt[3]{2}\). Since \(\sqrt[3]{2}\) is an algebraic number of degree 3 over the rationals, it is not a constructible number, making this construction impossible.

Answer:

• A. Doubling the square
• B. Trisecting a right angle
• C. Bisecting any angle
• D. Doubling the cube (Correct answer)