Questions: Trapezoid Properties

Question 1

A trapezoid has bases of 6 cm and 14 cm. What is the length of its midsegment?

• A. 20 cm — the sum of the two bases
• B. 10 cm — the average of the two bases
• C. 8 cm — the difference of the two bases
• D. 7 cm — half the longer base

Answer: (B)

The midsegment of a trapezoid equals the average (arithmetic mean) of the two bases: (6 + 14) / 2 = 10 cm. Option A confuses the formula with the perimeter contribution; option D takes half of only the longer base, ignoring the shorter one entirely. The average formula works because the midsegment sits exactly halfway between the two bases, balancing both extremes.

Question 2

A trapezoid is known to have congruent legs. Which additional properties are guaranteed?

• A. Congruent diagonals and congruent base angles
• B. Congruent diagonals only — base angles may differ
• C. Congruent base angles only — diagonals are unrelated to leg length
• D. Right angles at each vertex of one base

Answer: (A)

Congruent legs define an isosceles trapezoid, and symmetry cascades into two further properties: the two base angles sharing each base are congruent, and the diagonals are congruent. These all follow from the line of symmetry that runs through the midpoints of the two bases. Options B and C each grant only half the correct answer, and D is a property of right trapezoids, which are a different special case.

Question 3

Under the standard US definition, a parallelogram is a special case of a trapezoid.

• T. True
• F. False

Answer: False

The standard US definition of a trapezoid requires *exactly* one pair of parallel sides, which explicitly excludes parallelograms (which have two pairs of parallel sides). Under an alternative 'at least one pair' definition, parallelograms would be included — but that is not the conventional US classroom definition. Knowing which definition is in play matters because it changes whether squares and rectangles qualify as trapezoids.

Question 4

The area of a trapezoid equals the average of its two base lengths multiplied by its height.

• T. True
• F. False

Answer: True

The area formula A = ½(b₁ + b₂)h is exactly 'average of the bases times the height.' Thinking of it this way makes the formula intuitive: a rectangle with the same height and width equal to the average base would have the same area as the trapezoid, because the trapezoid is wider than the shorter base and narrower than the longer base — so the average is the right representative width.

Question 5

Why does the midsegment of a trapezoid equal the average of the two bases rather than the length of either base or some other value?

Model answer: The midsegment connects the midpoints of the two legs and sits exactly halfway between the two bases. Because it divides the trapezoid into two smaller trapezoids of equal height, it must be the value that perfectly balances the shorter and longer base — the arithmetic mean. Formally, using coordinates, if b₁ and b₂ are the base lengths at y = 0 and y = h, the midpoints of the legs lie at y = h/2, and the segment between them has length (b₁ + b₂)/2 by the midpoint formula.

The key insight is that 'halfway between' corresponds geometrically to the arithmetic average. This is the same logic as the midsegment theorem for triangles (midsegment = half the base), extended to the case where you have two parallel bases instead of one. Students who just memorize the formula miss the intuition: the midsegment is the 'middle' base, and the middle value between two numbers is their average.