Problem
Move a triangle onto its congruent twin
Square ABCD has side 6 cm. A point G on the top side joins to C, and a second square is built outwards on GC. Joining D to the new corner makes a triangle. Find the area of the shaded triangle.
Reasoning · Grade 5-2 Congruent Figures
Your answer
Watch out G's position seems essential, but the area is the same wherever it sits.
How to solve
Strategy Draw a Diagram — The shaded triangle DCE has no measurable side, so attacking it head-on is hopeless. Instead I change focus: I look for a different triangle in the picture that is the same shape and size and that does have measurable sides. Vertex C is the hinge of the whole figure — both squares have a corner there — so I try turning the picture a quarter turn about C and watch where things land. Tracing triangle BGC onto tracing paper, pinning the paper at C and giving it one quarter turn is enough to see the whole idea, and once I know the two triangles match I only need the base and height of the easy one. Trying G at a couple of convenient spots on AD then confirms that the answer really does not depend on where G is.
1STEP 1
Look for a twin of the shaded triangle
The squares give two pairs of equal sides.
CB = CD = 6 cm, CG = CE
A square hands you equal sides for free. Whenever a problem gives you two squares that share a corner, that corner is where the equal lengths pile up, so it is the first place to look for matching triangles.
6.G.A.1Change Focus Count The Complement2STEP 2
Show the two angles at C are equal
The two angles at C are equal too.
∠ BCG = 90° - ∠ GCD = ∠ DCE
Two angles that fill up the same right angle alongside the same third angle have to be the same size. You never need to know how big any of them is — that is the whole point, because the picture never tells you.
7.G.B.5Draw A Diagram3STEP 3
Turn triangle BGC a quarter turn about C
A quarter turn lands the shaded triangle onto its twin.
△ BGC → △ DCE → area(△ DCE) = area(△ BGC)
Turning a shape moves it without changing it, so area comes along for the ride. That is why swapping a hard-to-measure shape for the shape it turns into is always allowed.
8.G.A.2Create A Physical RepresentationTurning triangle BGC a quarter turn about C lands it on a triangle that is far easier to measure.
Why?
A turn lays the triangle onto a copy of itself, so its area travels along completely unchanged.
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
Why?
The quarter turn is exactly the angle between the two squares' sides at C, which is why the moved triangle lines up so neatly.
🧱Angle measure addsA ray drawn inside an angle cuts it into two smaller angles whose measures add back to the original, just as lengths do.
4STEP 4
Measure the easy triangle instead
The easy twin has area 18 cm².
area(△ BGC) = 6 × 6 ÷ 2 = 18 cm²
The height of a triangle is measured straight across to the base, not along a slanted side. Here the two sides BC and AD of the square are 6 cm apart everywhere, so the height is 6 cm without measuring anything.
6.G.A.1Draw A Diagram5STEP 5
Check that the position of G really does not matter
Moving G leaves the area unchanged.
area(△ BGC) = 1/2 × 6 × 6 = 18 cm² for every G on AD
Testing the two extreme positions is a cheap safety net: if the answer were going to move, moving G all the way to a corner would show it up straight away.
6.G.A.1Solve An Easier Related Problem6STEP 6
Confirm it straight from the scale drawing
So the shaded area is 18 cm².
area(△ DCE) = 6 × 6 ÷ 2 = 18 cm²
The quarter turn is exactly what makes this work: it swaps the two numbers, sending the point 6 up and 3 across from C to the point 6 across and 3 down from C, so the height of the new triangle is the base of the old one.
6.G.A.1Draw A DiagramAnswer
18 cm²
6 × 6 ÷ 2 = 18
The units are right: two lengths in centimetres multiplied and halved give square centimetres. The size is right too. The whole square ABCD covers 6 × 6 = 36 cm², and the shaded triangle came out as exactly half of that, 18 cm² — which is what you would expect for a triangle standing on a full 6 cm side of the square and reaching 6 cm away from it. The shaded triangle in the picture is clearly larger than a quarter of the square and clearly smaller than the square, so 18 cm² sits where it should. The strongest check is the one the problem is really about: the answer never used the length of GC or the position of G, which is why the problem can leave both unstated.
Takeaway
If a shape is impossible to measure, turn a shape you can measure onto it — a quarter turn about the shared corner moves triangle BGC exactly onto the shaded one, and turning never changes area.
- Look for a twin of the shaded triangle
- Show the two angles at C are equal
- Turn triangle BGC a quarter turn about C
- Measure the easy triangle instead
- Check that the position of G really does not matter
- Confirm it straight from the scale drawing