Problem
Equal-area triangles between parallel sides
Both diagonals are drawn in trapezoid ABCD. They cross at a point E. Pieces taken alone or joined make several triangles. Count the pairs of triangles with equal area.
Reasoning · Grade 5-1 Equal-Area Transformation (2)
Your answer
Watch out Measuring seems needed, but equal base and height settle it with no lengths at all.
How to solve
Strategy Make a Systematic List — A 'how many pairs' question is only safe once every triangle in the picture has been listed, so I first name all of them: four small ones round the crossing point and four big ones made of two small ones each. Then I use the one fact that needs no measurements - two triangles on the same base whose tips lie on a line parallel to that base have the same area - and I use it twice, once on the long parallel side and once on the short one. Chopping the common part off a pair of equal triangles gives a third pair for free. Finally I test the whole thing on one easy trapezoid with friendly numbers to make sure nothing else is accidentally equal.
1STEP 1
List every triangle in the picture
List the four small and four large triangles.
small: ABE, BEC, CDE, ADE large: ABC, BCD, ACD, ABD
Sorting the triangles into 'one piece' and 'two pieces glued' is a listing scheme that cannot skip anything, and it already shows how each big triangle is built out of small ones.
4.G.A.1Make A Systematic List2STEP 2
State the one fact that needs no lengths
Same base and height means same area.
area = (base) × (height) ÷ 2
Between two parallel lines the gap never changes, so sliding the tip of a triangle along the upper parallel line stretches it sideways but never changes its height - and so never changes its area.
6.G.A.1Draw A DiagramTwo triangles on the same base with their tips on a line parallel to it always have the same area, whatever they look like.
Why?
Sliding the tip along the parallel line never changes the height, so the area cannot change however slanted the triangle looks.
🧱Same base, tip on a parallel line, same areaSlide a triangle's tip along a line parallel to its base: the height never changes, so neither does the area, however slanted it looks.
Why?
The gap between two parallel lines is the same everywhere you measure it, which is exactly why the height stays put.
🧱Parallel lines keep a constant gapTwo parallel lines never meet and the distance between them is the same everywhere you measure it.
3STEP 3
Pair 1: take the long side BC as the base
Using the long side gives the first pair.
[ABC] = BC × h ÷ 2 = [BCD]
Same base, tips on a line parallel to it - the classic equal-area pair, and it works even though triangle ABC is tall and thin on the left while BCD leans to the right.
6.G.A.1Draw A Diagram4STEP 4
Pair 2: take the short side AD as the base
Using the short side gives the second pair.
[ABD] = AD × h ÷ 2 = [ACD]
Nothing about the first pair depended on which parallel side you started from, so the trick has to work a second time with the roles swapped.
6.G.A.1Draw A Diagram5STEP 5
Pair 3: take away the part the first pair shares
Removing the shared part gives the third pair.
[ABE] = [ABC] - [BEC] = [BCD] - [BEC] = [CDE]
Areas add and subtract like lengths of ribbon: cut the same piece off two equal pieces and what is left is still equal. Doing the same with pair 2 (removing the shared triangle ADE) gives the very same pair, not a new one.
3.MD.C.7Identify Subproblems6STEP 6
Check on an easy trapezoid whether anything else is equal
An easy example shows there are no more.
2, 3, 3, 4.5, 5, 5, 7.5, 7.5 → 3 equal pairs
The three pairs found by reasoning have to be equal in every trapezoid; a single numbered example is enough to show that the leftovers - ADE against BEC, or a small triangle against a big one - are not equal in general, so no fourth pair can be claimed.
6.G.A.1Solve An Easier Related Problem7STEP 7
Count the pairs
So there are 3 pairs.
[ABC] = [BCD], [ABD] = [ACD], [ABE] = [CDE] → 3 pairs
Every one of the three came from the same single fact about the parallel sides, so a trapezoid can be trusted to give three pairs and no more.
4.G.A.2Make A Systematic ListAnswer
3 pairs
No lengths were given, so the answer must not depend on any; each of the three pairs was justified only by 'same base, same height', which is true for every trapezoid, so the count is stable. The pieces also account for the whole figure: the four small triangles ADE, ABE, BEC, CDE together make up the trapezoid, and in the numbered check 2 + 3 + 4.5 + 3 = 12.5, the trapezoid's area. Nothing was double counted - ABE = CDE was obtained by subtracting the shared triangle rather than by naming a new one, and subtracting the other shared triangle ADE from pair 2 reproduces the same pair instead of a fourth. Finally, since AD is shorter than BC, triangle ADE has both a shorter base and a smaller height than BEC, so those two can never be equal, which is why the count stops at 3.
Takeaway
Slide the tip of a triangle along a line parallel to its base and the area never budges - that one idea finds all three equal pairs in the trapezoid.
- List every triangle in the picture
- State the one fact that needs no lengths
- Pair 1: take the long side BC as the base
- Pair 2: take the short side AD as the base
- Pair 3: take away the part the first pair shares
- Check on an easy trapezoid whether anything else is equal
- Count the pairs
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