Why Pythagoras Works Worksheets, Grade 8 (8.G.B.6) – Free PDF + Answer Key

First page of the why pythagoras works practice worksheet

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Watch: The Pythagorean Theorem Explained (a² + b² = c²)

Key idea

a² + b² = c², with c the hypotenuse. The square proof: (a + b)² − 2ab = a² + b² = c². Letters, not numbers, are what make it a proof. The converse — a² + b² = c² ⇒ right-angled — is a separate statement and it is the one that tests a triangle.

Sample problems

State the Pythagorean theorem, saying clearly which side c is.
Do the square count for legs 8 and 15: big square, four triangles, leftover. Check against the hypotenuse.
Do the square count for legs 5 and 5. What is the hypotenuse squared? What does that tell you about the hypotenuse itself?
Write the square proof in letters, showing each step of the algebra.

Common mistakes to watch for

Used 5×12 as the triangle's area, forgetting the half, giving 240 and a leftover of 49. The four triangles make two rectangles, so 2ab is the total — a useful check.
Stopped at 169 without connecting it to 13. The connection is the whole point of the count.
Said he needs to check more examples. That is the misconception rather than the fix — more cases still leave infinitely many unchecked. Also common: “he needs to explain it”, which is right and does not say what the explanation is.

The answer key explains each one, so you can tell what a wrong answer means.

How to use this set

Teach with the slides (about 15 minutes), or watch the video together.
Practice with the worksheet. Questions climb from fluency to application to reasoning.
Check understanding with the exit ticket, and send the homework home.
Mark with the answer key — each answer shows the steps and the most likely mistake.