Three squares and one equation. Find the square corner, name the legs and the hypotenuse, and then: square, add or subtract, take the root, and check that the answer makes sense. Why it is true, how to run it backward to test a corner, and where right triangles hide — ladders, screens, boxes, and the distance between two points. Study cards, hints, a practice quiz at three levels, and four workshop activities that check themselves.
A square corner, and a real square built on each short side: one holds 9, the other holds 16.
a² + b² = c². The squares add — and the last step is always the square root, because c² is an area and the question wants a length.
Four things people get wrong
⚠️People often think…
Legs of 3 and 4 give a hypotenuse of 7.
The sides do not add — the SQUARES do. 9 + 16 = 25, and √25 = 5. A hypotenuse of 7 would mean the triangle had no corner at all.
Square first, then add.
⚠️People often think…
Once you have 25, you are done.
25 is c² — the area of the square, not the length of the side. The side is √25 = 5. Forgetting the root is the single most common lost point in this unit.
The last step is always the square root.
⚠️People often think…
You always add the two numbers you are given.
Only if the missing side is the hypotenuse. If the unknown is a LEG, you subtract — and the answer must come out SHORTER than the hypotenuse, which is how you check yourself.
A leg can never be longer than the hypotenuse.
⚠️People often think…
It does not matter which side you call c.
c is always the longest side, sitting alone across from the square corner. Put a leg in c's spot and every number after that is wrong — including when you are testing whether a triangle is right at all.
c goes alone. Find the square corner first.
Square, combine, root, check
1Watch one
The legs are 3 and 4. Find the hypotenuse.
Square each leg: 9 and 16.
The unknown is the hypotenuse, so add: 25.
Take the root: √25 = 5.
c = 5 — longer than both legs, so it makes sense.
2Do one with me
The legs are 6 and 8. Find the hypotenuse.
6² =
8² =
√(36 + 64) =
💬One sentence, then you move on
Why do you have to take the square root at the end?
3Try one
The hypotenuse is 13 and one leg is 5. How long is the other leg?
I want a hint first
The 13 is already the hypotenuse, so the missing side is a leg — and a leg has to come out shorter. Subtract instead of adding.
💬Last one — then you're done here
How do you know whether to add or subtract?
Where this goes
Where this lives
A ladder leaning against a wall, the diagonal of a TV screen, and a carpenter checking a corner is square by measuring 3, 4 and 5.
What this feeds
That is the last unit of eighth grade. Algebra I picks it up from here.
Find one right angle near you and name the two sides that make it.
One card at a time — tap “Show me” to check yourself, then Next. Start at Foundation; when those feel easy, climb.
Helpful Hints
🪜 From a picture to a length — the ladder
Find the square corner → the two sides touching it are the legs, the one across is the hypotenuse → decide what is missing → hypotenuse: square, ADD, root · leg: square, SUBTRACT, root → sense check: longer than each leg, shorter than their sum.
Box, across, square, add or subtract, root, check.
🤝 The whole topic in one idea
The theorem is about three square AREAS: the two squares built on the legs cover exactly as much as the square built on the hypotenuse. Everything else — ladders, diagonals, distance between points — is finding the right triangle hiding in the picture.
⚠️ Traps the test loves
Adding the legs: 3 and 4 do not make 7. Square first.
Stopping at c². 25 is the area of the square; the side is √25 = 5.
Putting the hypotenuse in a leg’s spot. c goes ALONE.
Adding when you should subtract. If the unknown is a leg, the answer must be shorter than the hypotenuse.
Testing for a right triangle with the wrong side as c. Use the LONGEST side.
Squaring is not doubling. 6² is 36, not 12.
📐 Triples worth knowing — the chart
Triple
Check
Multiples you will meet
3, 4, 5
9 + 16 = 25
6-8-10 · 9-12-15 · 12-16-20 · 30-40-50
5, 12, 13
25 + 144 = 169
10-24-26
8, 15, 17
64 + 225 = 289
16-30-34
7, 24, 25
49 + 576 = 625
14-48-50
🎯 How the test will ask
A right triangle with one side missing — hypotenuse or leg? Decide before you calculate.
Three side lengths — “is this a right triangle?” (the converse).
A story: ladder, ramp, diagonal, walking two directions. Draw the triangle and mark the box.
Two points on a grid — “how far apart?” (across and up are the legs).
A box — “what is the longest rod that fits?” (use the theorem twice).
A picture of squares on the sides, or the four-triangle proof — explain why a² + b² = c².
✅ Before the test, can you…
Point to the legs and the hypotenuse in any right triangle, however it is turned?
Find a missing hypotenuse, and a missing leg?
Decide from three sides whether a triangle is right?
Find the right triangle in a word problem and sense-check your answer?
Find the distance between two points on the coordinate plane?
Explain, with the four triangles, why the theorem is true?
Pick your level
Look back at anything you missed — the hint that appeared is exactly what to reread tonight.
How sure did you feel?
Workshop
Work like a mathematician: name the parts of a turned triangle, decide whether to add or subtract, put a proof in order, then test whether a corner is really square. Every activity checks itself, and hints are free.
Your practice never leaves this device. There is no account and no sign-in. Your work is saved in this browser only, and you can erase it whenever you want.
Your practice record — saved on this device
This is your record of the module on screen — it stays here and goes nowhere. Independent means you got it right on the first tap; supported means you got it after the explain-and-retry, or marked ‘I had it’ on a revealed answer. Both count, and neither is a grade. If your teacher asks, copy the row or show them this screen.
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The answer key is for a teacher: it prints only from here, for the unit on screen. Print the study packet prints the study pages and a blank quiz — never the answers.