Question

You know a right triangle with legs of length 3 units and 4 units has a hypotenuse of length 5 units. Use this information and algebra to show that any right triangle with legs 3n and 4n has a hypotenuse of 5n. HINT: Start with n= 2, which results in side lengths of 6, 8, and 10. Show all of your algebraic steps

Answer 1

For any right-angled triangle, Pythagoras theorem holds.

Pythagoras theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Now, let us show that any right triangle with legs 3n and 4n has a hypotenuse of 5n

Let h represent the hypotenuse

to prove that h = 5n

Form Pythagoras theorem,

`\begin{gathered} h^2=(3n)^2+(4n)^2 \\ h^2=9n^2+16n^2 \\ h^2=25n^2 \\ h=\sqrt[]{25n^2} \\ h=\text{ 5n} \end{gathered}`

Since h = 5n, any right triangle with legs 3n and 4n has a hypotenuse of 5n

To show that this is always true, starting with n= 2, which results in side lengths of 6, 8, and 10

when n= 2

3n = 3(2)= 6

4n = 4(2) = 8

the hypotenuse will be 5n, that is 5(2) = 10

This can also be confirmed by using the Pythagoras theorem

`\begin{gathered} h^2=6^2+8^2 \\ h^2=36+64 \\ h^2=\text{ 100} \\ h=\sqrt[]{100} \\ h=10 \end{gathered}`

Hence , for any right triangle with legs 3n and 4n, the hypotenuse is always 5n