Question

A ladder is leaning against a wall. The top of the ladder is 9 feet above the ground. If the bottom of the ladder is moved 3 feet farther from the wall, the ladder will be lying flat on the ground, still touching the wall. How long, in feet, is the ladder?

Answer 1

So technically they are asking you to find the hypotenuse of a right angle triangle the height is 9ft and the length is 3ft so the formula is
A² +b²=c²

the a² and B² Being the height and length and label the third side c²

so..
9²+3²=c²
9×9+3×3=c²
81+9=c²
40=c²
√40=√c²
6.32=c
therefore the ladder is 6.32

well i think that is how you answer

Answer 2

Check the picture below

$\bf \textit{using the pythagorean theorem}\\\\ r^2=x^2+y^2\implies r=\sqrt{x^2+y^2} \\\\\\ r=\sqrt{x^2+9^2}\implies r=\sqrt{x^2+81}\impliedby \textit{leaning ladder} \\\\\\ r=\sqrt{(x+3)^2+0^2}\implies r=\sqrt{(x+3)^2}\impliedby \textit{flat ladder}\\\\ -------------------------------$

$\bf \sqrt{x^2+81}=\sqrt{(x+3)^2}\implies x^2+81=(x+3)^2 \\\\\\ x^2+81=x^2+6x+9\implies 81-9=6x\implies 72=6x\implies \cfrac{72}{6}=x \\\\\\ \boxed{12=x}\\\\ -------------------------------\\\\ \textit{now, the ladder "r" is }\sqrt{(x+3)^2}\implies \sqrt{(12+3)^2}\implies 15$