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The three sides of a right-angled triangle have lengths $a$, $b$ and $c$, where $a$, $b$, $c \in \mathbb{Z}$ - AQA - A-Level Maths: Pure - Question 6 - 2019 - Paper 3
Question 6
The three sides of a right-angled triangle have lengths $a$, $b$ and $c$, where $a$, $b$, $c \in \mathbb{Z}$. 6 (a) State an example where $a$, $b$ and $c$ are all ... show full transcript
Worked Solution & Example Answer:The three sides of a right-angled triangle have lengths $a$, $b$ and $c$, where $a$, $b$, $c \in \mathbb{Z}$ - AQA - A-Level Maths: Pure - Question 6 - 2019 - Paper 3
Step 1
Step 2
Prove that it is not possible for all $a$, $b$ and $c$ to be odd.
Answer
Assume that and are both odd. We can express them as:
- ,
for some integers and . According to the Pythagorean theorem, we have:
Substituting and gives:
Expanding this results in:
which simplifies to:
This can be rewritten as:
Since is even, it follows that must be even (as the square of an odd number is odd). This leads to a contradiction, as we started with the assumption that , , and are all odd. Thus, it is not possible for all of , , and to be odd.