Question $4$ ) Pythagorean triples are right angled triangles that have all sides as whole numbers. The smallest Pythagorean triple is $3,4,5$ . $\newline$ Show, using Pythagoras Theorem that this is a right angled triangle substituting the values in to the formula and showing that the left hand side of the equation equals the right hand side. $\newline$ Then find $3$ more Pythagorean triples and show them on triangles.

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Q. Question

4

) Pythagorean triples are right angled triangles that have all sides as whole numbers. The smallest Pythagorean triple is

3,4,5

\newline

Show, using Pythagoras Theorem that this is a right angled triangle substituting the values in to the formula and showing that the left hand side of the equation equals the right hand side.

\newline

Then find

3

more Pythagorean triples and show them on triangles.

Pythagoras Theorem Explanation: Pythagoras Theorem states that for a right-angled triangle, the square of the hypotenuse $c$ is equal to the sum of the squares of the other two sides $a$ and $b$ . The formula is $a^2 + b^2 = c^2$ .
Let's check if $3, 4, 5$ is a Pythagorean triple.
$a = 3$ , $b = 4$ , $c = 5$ .
Calculate $a^2 + b^2$ and compare it to $c^2$ .
$a$ $0$ .
Now, calculate $c^2$ .
$a$ $2$ .
Since $a$ $3$ , the left hand side equals the right hand side.
Checking $3$ , $4$ , $5$ Triple: Now let's find $3$ more Pythagorean triples. $\newline$ First, we can use multiples of the smallest Pythagorean triple, $3$ , $4$ , $5$ . $\newline$ Multiply each number by $2$ to get a new triple: $6$ , $8$ , $10$ . $\newline$ Check if it's a Pythagorean triple: $6^2 + 8^2 = 10^2$ . $\newline$ $36 + 64 = 100$ , and $10^2 = 100$ .
Finding More Triples: For the second new triple, multiply $3, 4, 5$ by $3$ : $9, 12, 15$ . $\newline$ Check if it's a Pythagorean triple: $9^2 + 12^2 = 15^2$ . $\newline$ $81 + 144 = 225$ , and $15^2 = 225$ .
Using Different Formula: For the third new triple, let's find a different set. We can use the formula $m^2 - n^2$ , $2mn$ , $m^2 + n^2$ where $m$ and $n$ are integers and $m > n$ . Choose $m = 3$ and $n = 2$ . Calculate $m^2 - n^2$ : $3^2 - 2^2 = 9 - 4 = 5$ . Calculate $2mn$ : $2mn$ $1$ . Calculate $m^2 + n^2$ : $2mn$ $3$ . The new triple is $2mn$ $4$ , $2mn$ $5$ , $2mn$ $6$ . Check if it's a Pythagorean triple: $2mn$ $7$ . $2mn$ $8$ , and $2mn$ $9$ .