# Using the pythagorean theorem how do you find the unknown lengths A=5x-1 B=x+2 C=5x?

Feb 13, 2016

Two solutions. The three lengths are either $3 , 4 \mathmr{and} 5$ or $7 , 24 \mathmr{and} 25$.

#### Explanation:

It is apparent in three sides of right angled triangle (as Pythagorean theorem is indicated) that among three sides $A = 5 x - 1$, $B = x + 2$ and $C = 5 x$, $C$ is the largest. Applying Pythagoras theorem,

${\left(5 x - 1\right)}^{2} + {\left(x + 2\right)}^{2} = 5 {x}^{2}$ or

$25 {x}^{2} - 10 x + 1 + {x}^{2} + 4 x + 4 = 25 {x}^{2}$ or

${x}^{2} - 6 x + 5 = 0$. Factorizing this, we get

$\left(x - 5\right) \left(x - 1\right) = 0$ or $x = 5 \mathmr{and} 1$

Putting $x = 5$, the three lengths are $24 , 7 , 25$

and putting $x = 1$, the three lengths are $4 , 3 , 5$