Question

the diagonals of any trapezoid are congruent.

a. true
b. false

Explanation:

Analyze the definition of a general trapezoid

A general trapezoid is a quadrilateral with at least one pair of parallel sides. The lengths of its non-parallel sides, and consequently its diagonals, do not have to be equal.

Identify counterexamples

Consider a right trapezoid with vertices at \((0,0)\), \((4,0)\), \((2,3)\), and \((0,3)\).
The parallel bases are along the x-axis and the line \(y = 3\).
The lengths of the diagonals are:

$$ d_1 = \sqrt{(2-0)^2 + (3-0)^2} = \sqrt{4 + 9} = \sqrt{13} $$

$$ d_2 = \sqrt{(0-4)^2 + (3-0)^2} = \sqrt{16 + 9} = 5 $$

Since \(\sqrt{13}
eq 5\), the diagonals are not congruent.

Evaluate the statement

The statement claims that the diagonals of any trapezoid are congruent. Since congruent diagonals are a specific property of isosceles trapezoids rather than all trapezoids, the statement is false.

Answer:

• A. True
• B. False (Correct answer)