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Geometry and Measurement 6-8 / Week 11 / Monday
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Week 11 · The Pythagorean Theorem

Monday

The ladder makes a triangle
// The brace: a² + b² = c² in every right triangle
⏱ about 20 min

Monday: The Ladder Makes a Triangle

Rocket leans a straw ladder against the model clubhouse wall. "The roof crew needs a way up. How long should it be?"

"What do you notice about the shape it makes?" Raven asks. She points at the wall, the floor and the straw.

"A triangle with a square corner where the wall meets the floor," Rocket says. "The ladder is the slanted side."

Raven measures twice. "The foot is 6 centimeters from the wall. The top is 8 centimeters up."

Nova hovers at the square corner, her light drawing a square on each side of the triangle. "Count the areas," she says.

Rocket sketches squares of 36 and 64 on the two short sides. "And the big one on the ladder?"

Nova hums. "Add the two small squares first. Then tell me the ladder."

Rocket, Raven and Nova at the Blueprint Table; a small ladder leans on the model clubhouse wall, making a right triangle.

Legs, hypotenuse and the squares

A right triangle has one square corner. The two sides that meet there are the legs, a and b.

The side across from the square corner is the hypotenuse, c. It is always the longest side.

Draw a square on each side. The two leg squares together have the same area as the hypotenuse square.

A right triangle with legs 3 and 4 and a square on each side, areas 9, 16 and 25

That is the Pythagorean theorem: a² + b² = c². With legs 3 and 4: 9 + 16 = 25, so c = 5.

The crew's ladder: legs 6 and 8. 36 + 64 = 100, and the square root of 100 is 10. The ladder is 10 cm.

A right triangle with legs 6 and 8 cm and the longest side marked ?

Finding the hypotenuse

  1. Square each leg: a × a and b × b.
  2. Add the two squares.
  3. Take the square root of the sum. That is c.
  4. Check: c must be longer than either leg, but shorter than the two legs added.
SQUARES ON THE LEGS
  • Read the question.
  • Tap your answer.
What is 3² + 4²?
What is 6² + 8²?
A square is drawn on a side 5 units long. What is its area, in square units?
What is 5² + 12²?
FIND THE LADDER
  • Read the question.
  • Tap your answer.
A right triangle with legs 3 and 4 cm and the longest side marked ?A right triangle has legs 3 and 4 cm. How long is the longest side, in centimeters?
A right triangle with legs 6 and 8 cm and the longest side marked ?The ladder's foot is 6 cm from the wall and its top is 8 cm up. How long is the ladder, in cm?
A right triangle with legs 5 and 12 cm and the longest side marked ?A right triangle has legs 5 and 12 cm. How long is the longest side, in centimeters?
A right triangle with legs 9 and 12 cm and the longest side marked ?A right triangle has legs 9 and 12 cm. How long is the longest side, in centimeters?
StatementTrue or false?
The hypotenuse is the longest side of a right triangle.?
The legs meet at the square corner.?
In a right triangle, a + b = c.?
The squares on the two legs together equal the square on the hypotenuse.?
WHY THIS EXERCISENaming the legs and the hypotenuse correctly is the first step in every problem this week.
A right triangle has legs 6 and 8. How long is the hypotenuse? Type the number.
WHY THIS EXERCISETwo legs are enough to find the third side, which is why a tape measure can skip the diagonal.
Try it
Lean a ruler against a box on the table so its foot is 6 cm from the box. Measure how high the top reaches.
Square both numbers, add, and take the square root. Then measure the ruler to check. No real ladders: look only.
Draw a right triangle with legs 3 and 4. Draw the square on each side and write each area inside it.

Two legs gave the ladder. Tomorrow four triangles in a square show why the rule always works.