Week 11 · The Pythagorean Theorem

Thursday

The brace log

// The brace: a² + b² = c² in every right triangle

⏱ about 20 min

Thursday: The Brace Log

Raven opens the Blueprint Book to the brace log. "Every model wall gets a diagonal brace, corner to corner."

"Six by eight wall, ten brace," Rocket reads. "Nine by twelve, fifteen. The straws are cut already."

"What do you notice about the last wall?" Raven asks. "Seven by nine. The brace is not a whole number."

Rocket squares and adds. "Forty-nine plus eighty-one is 130. The square root is about 11.4."

Nova hovers above the storage box, her light shining into one corner. "A straw must lie flat inside," she says.

"Corner to far corner, through the box?" Rocket asks. "That is two triangles. Floor first, then up."

Nova hums. "Floor diagonal, then height. Measure twice before you cut."

Braces, flat and through a box

A brace runs corner to corner across a rectangular wall. The wall's width and height are the legs, and the brace is the hypotenuse.

When the sum of squares is not a perfect square, give the root to one decimal place. √130 is about 11.4.

For a diagonal through a box, go in two steps. First the floor diagonal, then that diagonal and the height.

These are the crew's own Blueprint Table measurements for the model clubhouse walls, in centimeters.

Wall width	Wall height	Brace length
6	8	10
9	12	15
5	12	13
8	15	17
7	9	11.4

$A right triangle with legs 7 and 9 cm and the longest side marked ?$

The storage box is 3 by 4 by 12 cm. Floor diagonal: 3² + 4² = 25, so 5.

Then 5² + 12² = 169, and the square root is 13. A 13 cm straw just fits.

READ THE BRACE LOG

Read the question.
Tap your answer.

$A right triangle with legs 9 and 12 cm and the longest side marked ?$ A wall is 9 cm wide and 12 cm tall. How long is its brace, in cm?

$A right triangle with legs 8 and 15 cm and the longest side marked ?$ A wall is 8 cm wide and 15 cm tall. How long is its brace, in cm?

$A right triangle with legs 7 and 9 cm and the longest side marked ?$ A wall is 7 cm wide and 9 cm tall. How long is its brace, to one decimal place?

$A right triangle with legs 12 and 5 cm and the upright leg marked ?$ A brace is 13 cm long on a wall 12 cm tall. How wide is the wall, in cm?

THROUGH THE BOX

Read the question.
Tap your answer.

$A right triangle with legs 3 and 4 cm and the longest side marked ?$ The box floor is 3 by 4 cm. How long is the floor diagonal, in cm?

A box is 3 by 4 by 12 cm. How long is the diagonal from one corner to the opposite corner, in cm?

A smaller box is 2 by 3 by 6 cm. How long is the diagonal through it, in cm?

The 6 by 8 wall: how long is its brace, in cm? Type the number.

The 7 by 9 wall: what is 7² + 9²? Type the number.

The longest straw that fits through the 3 by 4 by 12 box, in cm? Type the number.

Statement	True or false?
The brace of a rectangular wall is the hypotenuse of a right triangle.	?
A root that is not whole is written to one decimal place in this course.	?
A diagonal through a box needs only one use of the theorem.	?
Knowing the brace and the height, you can find the width.	?

WHY THIS EXERCISEReal builds need lengths that are not whole numbers, and two-step diagonals are common.

Try it

Measure the width and height of a cardboard box side with a tape measure. Compute its diagonal.
Then measure the diagonal corner to corner. How close was the computed length?

Draw the 3 by 4 by 12 box. Draw the floor diagonal, then the long diagonal from a floor corner to the top far corner.

Blueprint Book: copy the brace log and add a wall of your own with its brace length.Write the two steps for a diagonal through a box, and why the first answer becomes a leg.Rocket said the straws were cut already. Write why computing before cutting saves cardboard and straws.

Braces and box diagonals done. Tomorrow the theorem measures the grid and checks a corner.

Grown-up checked my answers

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