1. ## pythagoras

the lengths, in centimetres, of the sides of a right-angled triangle are:
x, x+2 and x+5

i) use pythagoras' theorem to write down an equation in x. show that it simplifies to
x^2-6x-21=0
ii) solve the equation

iii) write down the length of hypotenuse of the triangle.

2. Hello, simonmann!

The problem is straight-foward.

The lengths of the sides (in cm) of a right triangle are: $\displaystyle x,\:x+2,\:x+5$

a) Use Pythagoras' Theorem to write an equation in $\displaystyle x.$
$\displaystyle a^2 + b^2 \:=\:c^2 \quad\Rightarrow\quad x^2 + (x+2)^2 \:=\:(x+5)^2$

Show that it simplifies to : .$\displaystyle x^2-6x-21\:=\:0$
We have: .$\displaystyle x^2 + x^2 + 4x + 4 \;=\;x^2 + 10x + 25\quad\Rightarrow\quad x^2-6x-21 \:=\:0$

b) Solve the equation.
Quadratic Formula: .$\displaystyle x \;=\;\frac{-(\text{-}6) \pm\sqrt{(\text{-}6)^2 - 4(1)(\text{-}21)}}{2(1)} \;=\;\frac{6\pm\sqrt{120}}{2} \;=\;3 \pm\sqrt{30}$

And the positive root is: .$\displaystyle x \:=\:3 + \sqrt{30}$ cm.

c) Find the length of hypotenuse of the triangle.
The hypotenuse is: .$\displaystyle x + 5 \;=\;(3 + \sqrt{30}) + 5 \;=\;8 + \sqrt{30}$ cm.