The two quadrilaterals that are not parallelograms, and the shortcut that connects midpoints.
Find the median
A trapezoid has parallel sides of 14 cm and 22 cm. How long is its median (midsegment)?
The median is the average of the two parallel sides: 14 + 222 = 18 cm. C forgets to halve, and A subtracts. A sanity check settles it instantly — the median runs between the two sides, so its length must lie between 14 and 22.
| Part | Which sides |
|---|---|
| Bases | the two parallel ones |
| Legs | the two non-parallel ones |
| Median | joins the midpoints of the legs |
"Exactly one pair" is what separates a trapezoid from a parallelogram. Two pairs of parallel sides would make it a parallelogram instead.
m = b₁ + b₂2
bases 14 and 22
find the median
14 + 222 = 18
between the two ✓
median 15, one base 11
find the other base
11 + b2 = 15
set it up
b = 19
11 + b = 30
The area formula uses the same average: A = (b₁ + b₂)h2 is just median × height.
| Property | Detail |
|---|---|
| Legs equal | the definition |
| Base angles equal | the two at each base match |
| Diagonals equal | though they do not bisect each other |
| Opposite angles | supplementary |
The diagonals of an isosceles trapezoid are the same length, but they do not cut each other in half. That last part is a parallelogram property, and a trapezoid is not one.
Area of a kite = d₁ × d₂2, half the product of the diagonals — the same formula a rhombus uses, since a rhombus is a kite with all four sides equal.
The segment joining two midpoints is parallel to the third side and half its length.
In triangle ABC, BC = 26
D and E are midpoints of AB and AC
DE = 26 ÷ 2 = 13
half the third side
DE is parallel to BC
and that comes free
This is the trapezoid median in miniature. A triangle is a trapezoid whose top base has shrunk to nothing, and 0 + 262 = 13 gives the same answer.
When one base is described from the other, call the shorter one b and write the longer with b. Angles on the same leg lie between the parallel bases, so they add to 180°.
A trapezoid has a median of 15, and one base is 4 longer than the other. Find both bases.
bases b and b + 4
one letter for both
b + b + 42 = 15
the median is the average
2b + 4 = 30, so b = 13
double, then solve
bases 13 and 17
check: 13 + 172 = 15 ✓
Writing b + (b + 4) = 15 treats the median as the sum of the bases. It gives b = 5.5 and bases of 5.5 and 9.5 — whose median is only 7.5. The median is the sum divided by 2.
An isosceles trapezoid has a base angle of 70°. Find all four angles.
70° and 70°
base angles are equal
180° − 70° = 110°
same leg: supplementary
70°, 70°, 110°, 110°
check: total 360° ✓
The same facts, with the unknown moved to a new place.
D and E are midpoints of AB and AC; DE = x + 4 and BC = 3x − 2. Find BC.
2(x + 4) = 3x − 2
the third side is twice the midline
2x + 8 = 3x − 2, so x = 10
solve
BC = 3(10) − 2 = 28
check: DE = 14, half of 28 ✓
Double the midline, not the side: the "2" goes on the shorter segment.
In a kite, the angles where the equal sides meet are 110° and 50°. Find the other two.
360° − 110° − 50° = 200°
angles of a quadrilateral
200° ÷ 2 = 100° each
the remaining pair are equal
A kite has one pair of equal opposite angles — the two between the unequal sides.
A trapezoid has area 84 cm² and bases 9 cm and 15 cm. Find its height.
median = 9 + 152 = 12
area = median × height
12h = 84, so h = 7 cm
check: 12 × 7 = 84 ✓
Running the area formula backwards: divide the area by the median.
Expect a trapezoid median from two bases or a base from the median, the midline of a triangle, a "which property belongs to which shape" item, and an angle along a leg of an isosceles trapezoid. The fastest route is the average — median = b₁ + b₂2, so double the median to get the sum of the bases — and the midline is half the third side. The 10-second check: a median must fall between the two bases, and the angles on one leg add to 180°.
Find the base
A trapezoid has a median of 17 cm and one base of 12 cm. How long is the other base?
The median averages the bases: 12 + b2 = 17, so 12 + b = 34 and b = 22 cm. Check: 12 + 222 = 17 ✓ B subtracts and D forgets to subtract the 12 after doubling. Sanity check: the median of 17 sits between 12 and 22, as it must.
Hint: double the median first to get the sum of both bases, then subtract the one you know.
