# Using the pythagorean theorem how do you find the unknown lengths A=x B=2x-3 C=2x+3?

$a = 24$ and $b = 45$ and $c = 51$

#### Explanation:

From Pythagorean Theorem, we know that
${c}^{2} = {a}^{2} + {b}^{2}$ where a, b are the sides or legs
and c the hypotenuse

From the given,
$c = 2 x + 3$
$a = x$
$b = 2 x - 3$

${c}^{2} = {a}^{2} + {b}^{2}$
direct substitution

${\left(2 x + 3\right)}^{2} = {x}^{2} + {\left(2 x - 3\right)}^{2}$

expand the equation

$4 {x}^{2} + 12 x + 9 = {x}^{2} + 4 {x}^{2} - 12 x + 9$

simplify

${x}^{2} - 24 x = 0$

solution by factoring

$x \left(x - 24\right) = 0$
the roots are

$x = 0$ and $x = 24$

with $x = 0$ the sides are $a = 0$, $b = - 3$ and $c = 3$
with $x = 24$ the sides are $a = 24$, $b = 45$ , $c = 51$

Check:
${c}^{2} = {a}^{2} + {b}^{2}$
${51}^{2} = {24}^{2} + {45}^{2}$
$2601 = 576 + 2025$
$2601 = 2601$
correct !!

God bless...I hope the explanation is useful.