Right triangles. Exact relationships.
Find a missing side, calculate the distance between two points, or explore why the three square areas belong in one equation. Print focused practice with separate worked solutions.
Choose a worksheetSee the relation. Show your work.
Choose a four-question practice sheet with diagrams, room to show work and a separate worked answer key.
Pythagorean theorem
Print a four-question Pythagorean theorem worksheet for missing hypotenuses or legs. Choose whole-number triples or irrational results from whole or decimal givens.
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Pythagorean theorem
Find distances on four coordinate grids with whole-number, decimal-tenths, radical or mixed coordinates. Choose first-quadrant or four-quadrant points and print separate worked solutions.
Preview and downloadSee the relation. Show your work.
Use six illustrated pages to connect the equation to square areas and follow each kind of calculation.
Pythagorean theorem
A six-page printable guide to right-triangle parts, square areas, missing sides, irrational lengths and the coordinate distance formula.
Preview and downloadThe hypotenuse is the side opposite the right angle. Its square equals the sum of the two leg squares. When the hypotenuse is missing, add the leg squares; when a leg is missing, subtract the known leg square from the hypotenuse square.
For legs 5 and 12, the sum is 25 + 144 = 169, so the hypotenuse is 13. With hypotenuse 13 and leg 5, the difference is 169 − 25 = 144, so the other leg is 12.
Whole-number triples let learners verify each answer without a calculator. Next, keep the givens whole while introducing a noninteger square root. Finally, use given lengths with tenths to practice squaring decimals accurately.
The answer key keeps intermediate squares exact. A noninteger length is rounded only at the last step, to four decimal places, and is marked as an approximation.
Subtract the x coordinates to find horizontal change and the y coordinates to find vertical change. These changes form the two perpendicular legs; the straight line between the points is the hypotenuse.
From (−2, 1) to (1, 5), the changes are 3 and 4. Their squared sum is 25, giving a distance of 5 units. Either point can come first as long as the order stays consistent in both differences.
Choose by readiness
These are enrichment entry points for learners ready for squares, roots and coordinate differences. Grade labels help navigation; they do not claim alignment to a country or curriculum. Many students study these skills later than Grade 6.
Begin with the illustrated explanation after learners know square area and right angles.
Open this starting point →An enrichment starting point for learners who can square numbers and recognize a positive square root.
Open this starting point →Mix hypotenuse and leg questions after learners can rearrange the squared-length relation.
Open this starting point →Connect coordinate differences to the two legs of a right triangle; introduce signed grids later.
Open this starting point →Prepare for the next step
Graphing
Practice signed coordinates with 24 reading and plotting questions. Identify labeled points, write ordered pairs and add new points across the four quadrants.
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Geometry
Decide whether six pairs of extended supporting lines are perpendicular. True examples include a right-angle mark; counterexamples include parallel and oblique pairs.
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Graph paper
Print first-quadrant or four-quadrant coordinate planes in seven layouts. Choose numbered or unnumbered axes while keeping the declared coordinate ranges.
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