Answer:
[tex]\sf x = 1 + 2\sqrt{3}[/tex]
Explanation:
In a right angle triangle, the hypotenuse is the longest side.
Here the given sides are (x + 1) cm, (x + 2) cm, (x + 4) cm.
where (x + 4) is the longest side among all the side lengths.
Apply Pythagoras theorem:
(leg 1)² + (leg 2)² = (hypotenuse)²
substitute values
[tex]\sf (x + 1)^2 + (x + 2)^2 = (x + 4)^2[/tex]
[tex]\sf x^2 + 2x + 1 + x^2 + 4x + 4 = x^2 + 8x + 16[/tex]
[tex]\sf x^2 + x^2 - x^2 + 2x + 4x - 8x + 1 + 4 - 16 = 0[/tex]
[tex]\sf x^2 - 2x - 11 = 0[/tex]
use quadratic formula
[tex]\sf \dfrac{-(-2)\pm \sqrt{(-2)^2 - 4(1)(-11)} }{2(1)}[/tex]
[tex]\sf \rightarrow \dfrac{2\pm \sqrt{48} }{2}[/tex]
[tex]\sf \rightarrow 1\pm \sqrt{48}[/tex]
[tex]\sf \rightarrow 1\pm 2\sqrt{3}[/tex]
[tex]\sf \rightarrow 1+ 2\sqrt{3} \:\ or \:\ 1 -2\sqrt{3}[/tex]
[tex]\sf \rightarrow 4.46 \:\ or\:\ -2.46[/tex]
As length can only be positive, the value of x will be 1 + 2√3.
