Video: AQA GCSE Mathematics Higher Tier Pack 5 β’ Paper 2 β’ Question 24

A) The converse to Pythagorasβs theorem states that a triangle with lengths π, π, and π, where π is the hypotenuse, is a right-angled triangle if πΒ² + πΒ² = πΒ². For example, a triangle with lengths 3, 4, and 5 must be right-angled because 3Β² + 4Β² = 5Β². Prove that the triangle with lengths 50.1, 120, and 130.1 is not right-angled. B) A Pythagorean triple is a set of three positive integers that form a right-angled triangle. For example, 3, 4, 5 is a Pythagorean triple. Michael notices that 3π, 4π, 5π for any positive integer π is also a Pythagorean triple. Joe says that the same is true for 3 + π, 4 + π, and 5 + π. Find a counterexample to show that Joe is wrong.

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Video Transcript

The converse to Pythagorasβs theorem states that a triangle with lengths π, π, and π, where π is the hypotenuse, is a right-angled triangle if π squared plus π squared equals π squared. For example, a triangle with lengths three, four, and five must be right-angled because three squared plus four squared equals five squared. Prove that the triangle with lengths 50.1, 120, and 130.1 is not right-angled.

Part b) A Pythagorean triple is a set of three positive integers that form a right-angled triangle. For example, three, four, five is a Pythagorean triple. Michael notices that three π, four π, five π for any positive integer π is also a Pythagorean triple. Joe says that the same is true for three plus π, four plus π, and five plus π. Find a counterexample to show that Joe is wrong.

Part a) If the triangle is right-angled, it will satisfy Pythagorasβs theorem, as the example given β three, four, five β does. We need to show then if we substitute these values into the equation for Pythagorasβs theorem, the values do not satisfy the equation.

Remember, the hypotenuse is the longest side. So π is 130.1 in this case. Letβs begin by squaring the values of π and π. They are 50.1 and 120. 50.1 squared plus 120 squared is equal to 16910.01. Next, weβre going to square the value of π, which we said was 130.1. Thatβs 16926.01. The value for π squared plus π squared is not equal to the value for π squared. So this triangle cannot be right-angled.

For part b), the clue here is in the word βcounterexample.β A counterexample is any single example that proves the statement is incorrect. Remember, to prove something is true, itβs important to show that itβs true for every single value. But to disprove a statement, we just need to find one example where itβs not true.

A sensible starting choice here is to choose a low value of π. Letβs try one. Then three plus π is equal to four, four plus π is equal to five, and five plus π is equal to six. Once again, the longest side is the hypotenuse. So six is equal to π.

Weβre going to work out the values of π squared plus π squared. Here thatβs four squared plus five squared, which is equal to 41. π squared is six squared, and thatβs 36. We can say that 41 is not equal to 36. And weβve shown that Joe is wrong. When π is equal to one, three plus π, four plus π, and five plus π do not form a Pythagorean triple.