Prove the Pythagorean relation and use its converse without assuming the desired right angle.
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A right triangle with legs 3 and 4 has hypotenuse 5 because 3²+4²=25. The relation concerns squares of lengths, not a sum of raw lengths. The longest side, opposite the right angle, is the hypotenuse.
Predict
Predict which hypotheses permit the equation a²+b²=c². The forward theorem requires a right triangle and assigns c to its hypotenuse. The converse starts with a triangle whose side squares satisfy the equation and concludes a right angle. These are different arguments.
Experiment
Inspect the area dissection. The outer square has side 7 and area 49. Four 3–4 right triangles each occupy area 6, leaving 25 in the middle. Trace each central side: it is a triangle’s hypotenuse, so all four have length 5. At every central corner, the adjacent acute angles together are 90°, leaving a right angle.
Pythagorean dissection for a=3, b=4 and c=5, drawn to one scale. Four congruent right triangles surround a central square whose sides are their hypotenuses.
Understand
Worked argument: for general positive legs a,b, the same arrangement has outer area (a+b)² and four triangle areas totaling 2ab. The central square has area c². Therefore (a+b)²=2ab+c², and expanding yields a²+b²=c². The square justification matters; four equal sides alone could make a non-square rhombus.
For the converse, construct a separate right triangle with legs a,b. Its hypotenuse has the required length c by the forward theorem, so SSS transfers its right angle to the original triangle. Invalid proof: using Pythagoras on the original triangle before proving it right assumes the desired conclusion.
Apply
Name the hypotenuse before substituting. If a leg is unknown, subtract its companion square from the hypotenuse square. Use the nonnegative square root for a length. The exact checker accepts rational values; arguments involving irrational lengths should keep them symbolic in written work.
Continue
Depth: with fixed adjacent sides a,b, increasing their included angle increases the opposite side. Comparing against a right triangle therefore shows that c² greater than a²+b² indicates an obtuse opposite angle, and a smaller square indicates an acute one. This comparison is later quantified by the law of cosines.
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Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
A right triangle has legs 9 and 12. Find its hypotenuse.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A right triangle has hypotenuse 17 and one leg 8. Find the other leg.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Which side triple forms a right triangle?
A triangle has sides 5,6,8. Its largest angle is:
Prove a²+b²=c² using a square of side a+b tiled by four congruent right triangles with legs a,b and a central square of side c.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
If a triangle has positive side lengths a,b,c with a²+b²=c², prove the angle opposite c is right by constructing a comparison triangle.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
