Explain a proof of the Pythagorean Theorem and its converse.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
The Pythagorean Theorem says that in any right triangle, the squares built on the two legs together have the same area as the square built on the hypotenuse: a² + b² = c². Eighth graders do not just use this fact, they explain why it is true. A favorite argument arranges four copies of the right triangle inside a large square of side a + b. The space left over is a tilted square of area c². Rearranging the same four triangles leaves two squares of areas a² and b² instead. Since the big square and the triangles have not changed, the leftover areas must be equal.
The converse runs the other way: if the side lengths of a triangle satisfy a² + b² = c², then the triangle must have a right angle opposite the longest side. That gives a numerical test for squareness. A triangle with sides 9, 12 and 15 passes (81 + 144 = 225), so it is right-angled, while one with sides 6, 7 and 9 fails (36 + 49 = 85, not 81). Students should be able to explain one proof in their own words and to apply the converse confidently.
The hypotenuse is always the longest side, opposite the right angle. Testing 5, 12, 13 as 13² + 5² = 12² gives a false result.
Some students test 3, 4, 5 by checking whether 3 + 4 equals 5. The theorem is about areas, so each length must be squared first: 9 + 16 = 25.
a² + b² = c² holds only for right triangles. For an acute or obtuse triangle the two sides of the equation differ.
Checking a few triangles is evidence, not proof. The rearrangement argument works for every right triangle at once.
A triangle has sides 8 cm, 15 cm and 17 cm. Is it a right triangle?
Answer: Yes. 8² + 15² = 289 = 17², so it is a right triangle.
Have students cut out four congruent right triangles and physically arrange them in the two configurations inside a square frame. Talking through what changed and what did not is the proof. Afterwards, ask them to write it in three or four sentences, which is the level of explanation the standard expects.
For the converse, a builder's check works well as a story: a 3-4-5 rope triangle makes a square corner. Assessment items typically give three side lengths and ask whether the triangle is right, so emphasize identifying the longest side first.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: D) 5, 12, 13
5² + 12² = 25 + 144 = 169 = 13², so the converse says the triangle is right-angled. The other sets fail the test.
Answer: 225
c² = 9² + 12² = 81 + 144 = 225, so c = 15.
Answer: B) Because the big square and the four triangles have the same total area in both arrangements
The outer square and the four triangles do not change, so whatever area remains must be the same in both arrangements: c² in one and a² + b² in the other.
Answer: 85
36 + 49 = 85. Since 9² = 81 and 85 is not 81, the triangle is not right-angled.
Answer: D) The side opposite the right angle
The hypotenuse is opposite the right angle and is always the longest side.
A full lesson with slides, activities and an exit ticket on proving the pythagorean theorem and its converse, pitched to grade 8 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 8.G.B.6 with an answer key, ready in about a minute.
Make a worksheet →Turn proving the pythagorean theorem and its converse into a quiz students answer online that marks itself, with a class summary for you.
Build a test →They need to explain a proof, usually a visual or area argument, in their own words. They do not need to invent a new proof or write a formal one.
It tests whether a triangle is right-angled from its side lengths alone: if a² + b² = c² with c the longest side, the triangle has a right angle.