Week 11 · The Pythagorean Theorem
Build Day
Build Day: the string corner check
// The brace: a² + b² = c² in every right triangle
⏱ about 20 min
Build Day: The String Corner Check
Saturday at the Blueprint Table, Rocket holds a loop of string with knots at 30, 70 and 120 centimeters.
"Three, four, five, times ten," he says. "Pin the first knot at the corner, pull the sides tight, and look."
Raven pins the string along the two floor edges of the model base. "What do you notice at the third knot?"
"It reaches exactly. The far knots meet," Rocket says. "So the corner is square. The converse says so."
Nova hovers above the loop, her light tracing the triangle. "Thirty squared plus forty squared," she says.
"Nine hundred plus sixteen hundred. Twenty-five hundred. Fifty squared," Rocket says. Nova hums. "Square corner, square base."
- A right triangle has two legs at the square corner and a hypotenuse, the longest side.
- a² + b² = c²: four triangles rearranged in a square prove it for every right triangle.
- Find the hypotenuse by squaring, adding and taking the root. Find a leg by subtracting the squares.
- The distance between grid points is the hypotenuse of a run-and-rise triangle. A box diagonal takes two steps.
- The converse checks a corner: if a² + b² = c², the corner is square. Builders use 3, 4 and 5.
◇ THE 3-4-5 CORNER CHECK WITH A GROWN-UP
Tie knots in a string at 0, 30, 70 and 120 cm. The three sides will be 30, 40 and 50.
Pin or hold the first knot at a corner: a table, a box, a doorway on the floor.
Pull one side along each edge. If the 50 cm side reaches exactly, the corner is square.
Try a corner that is not square, like a folded piece of cardboard. Does the string fit?
Pin or hold the first knot at a corner: a table, a box, a doorway on the floor.
Pull one side along each edge. If the 50 cm side reaches exactly, the corner is square.
Try a corner that is not square, like a folded piece of cardboard. Does the string fit?
For a grown-up
This week your student learned the Pythagorean theorem, its proof, its converse and the distance between grid points.
Ask them why 30, 40 and 50 prove a square corner. The answer should mention 900 + 1600 = 2500.
Ladders are only looked at, never climbed. The string stays on the table or floor, never around anyone.
Ask them why 30, 40 and 50 prove a square corner. The answer should mention 900 + 1600 = 2500.
Ladders are only looked at, never climbed. The string stays on the table or floor, never around anyone.
Safety first
Keep the string flat on the table or floor. Never loop it around a neck, wrist or pet.
Pins go into cardboard only, with a grown-up nearby. Count them back into the box when done.
Pins go into cardboard only, with a grown-up nearby. Count them back into the box when done.
BUILD DAY CHECK
- Read the question.
- Tap your answer.
The string sides along the edges are 30 and 40 cm. How long must the third side be, in cm?A triangle has sides 30, 40 and 50 cm. Is its corner square?
Rocket pulls the string and the third side comes out 55 cm. Sides 30, 40 and 55: is the corner square?
The string loop has sides 50 and 30 cm at a square corner. How long is the other side, in cm?Draw the model base from above with the 30-40-50 string triangle pinned in one corner.