The converse of Pythagoras' theorem and Pythagorean triples
Mathematics Prep 2 Term 1, Unit 3, Lesson 2
GoalTest whether three lengths make a right-angled triangle, and recognise Pythagorean triples.
New words
converse · if c² = a² + b², then the triangle is right-angled opposite cPythagorean triple · three whole numbers with c² = a² + b², such as 3, 4, 5longest side · always the one used as c in the testmultiples · multiplying a triple by a number gives another triple
Explanation
The converse tells us when a triangle has a right angle: if the square of the longest side equals the sum of the squares of the other two, the angle opposite the longest side is a right angle. Why? Build a right-angled triangle with legs a and b. By the theorem its hypotenuse is c. A triangle with sides a, b and c is then congruent to it by SSS, so it must also have a right angle. To test 7, 24, 25: 7² + 24² = 49 + 576 = 625 = 25², so yes. To test 8, 13, 15: 8² + 13² = 233, but 15² = 225, so no. Always put the LONGEST side as c, and square all three. Triples such as 3-4-5, 5-12-13, 8-15-17 and 7-24-25 are Pythagorean triples, and their multiples, such as 6-8-10, are too. Builders use 3-4-5 to make square corners.
Triple
a² + b²
c²
3, 4, 5
9 + 16 = 25
25
5, 12, 13
25 + 144 = 169
169
8, 15, 17
64 + 225 = 289
289
7, 24, 25
49 + 576 = 625
625
Worked examples
Is a triangle with sides 20, 21 and 29 right-angled? The longest side is 29. 20² + 21² = 400 + 441 = 841 and 29² = 841. They are equal, so yes: the right angle is opposite the side 29.
Is 6, 7, 10 a right-angled triangle? 6² + 7² = 36 + 49 = 85, but 10² = 100. They are different, so no. (The triangle exists, since 6 + 7 > 10, but it has no right angle.)
A builder marks 3 m along one wall from a corner and 4 m along the other. If the distance between the two marks is exactly 5 m, then 3² + 4² = 5² and the corner is a right angle. A 6-8-10 or a 9-12-15 triangle works in the same way.
Class activity
Find a triple. Choose the 3-4-5 triple and multiply by 2, 3 and 10. Check each result with c² = a² + b². Then try the numbers 2, 3, 4 and show they do not make a right angle.
Practice
1
Is 5, 12, 13 a right-angled triangle?
25 + 144 = 169 = 13². Yes.
2
Is 8, 9, 12 a right-angled triangle?
64 + 81 = 145, but 12² = 144. They differ by 1, so no.
3
Find c so that 8, 15, c is a Pythagorean triple.
64 + 225 = 289, so c = 17.
4
Is a triangle with sides 1.5, 2 and 2.5 right-angled?
2.25 + 4 = 6.25 = 2.5². Yes.
5
Seif tests 7, 24, 25 by 7² + 25² = 24² and says ‘no’. What went wrong?
He did not use the longest side as c. The test is 7² + 24² = 25², which is true.
