Euler's Eyes
2026-10-21
Triangles that grow without changing shape
A comic-drawn ramp
On a mesh, draw a path 3 comics to the right and 4 comics to the top. Connect the beginning to the end by a segment. You have formed a triangle with a straight corner: a right triangle.
Now make a copy with all lengths folded. The horizontal path will have 6 comics; the vertical, 8. The lengths changed, but the corner openings remained the same.
$$3\cdot2=6,\qquad4\cdot2=8.$$Why does bending just one side change the shape?
Compare two copies: in the first, fold the horizontal and vertical path; in the second, fold only the horizontal. In the second copy the ramp is more laid down. Growing without changing shape requires expanding in all directions in the same proportion.
This experience prepares an idea that will be demonstrated in Fundamental II: triangles with the same angles have corresponding proportional sides.
The two equal sides of a fold
Fold a square diagonally. Both sides of the straight corner were the same length before bending and remain the same. The two smaller corners are also the same. The drawing is a model of a right-angled triangle with two equal sides.
If one of these sides measures 5 cm, the other also measures 5 cm:
$$a=b=5\text{ cm}.$$The letter is just a name for the measurement. The diagonal is another side; the above equality does not say how much it measures.
In everyday life
- Enlarged photocopy. Enlarging a figure on the copier enlarges all sides by the same proportion: the shape does not change.
- Maps. A map is a reduced copy of the place; Therefore, the angles between the streets are the same as those on the map for the city.
- Models. Architects make models with all measurements divided by the same number, and the roofs have the same slope.
Investigate with paper
Draw two triangles made by the diagonal of squares of different sizes. Place the smaller corners on top of each other. The directions coincide even if a triangle is larger. Later we will use this property to measure heights and distances.
Exercises with resolution
1. A copy triples all sides of a triangle. One side of 4 cm now measures how much?
Check resolution 1
$3\cdot4=12$ cm.
2. A diagonal divides a square of side 7 cm. How long are the two sides that form the right corner of each triangle?
Check resolution 2
Each measures 7 cm. The length of the diagonal will be studied with Pythagoras.
Correspondence with source material
Preparatory activity on angles and triangles, adapted for this stage. General formulas appear in the following tracks.
The text of this class is a didactic rewrite. The page transcription preserves the sequence of exercises and other items from the original; it still has OCR snippets pending review.