# Question 22db6

Mar 8, 2016

I found $5$

#### Explanation:

I would call the longer leg $x$ so that the shorter becomes $x - 7$.
Using Pythagoras we get:
${13}^{2} = {x}^{2} + {\left(x - 7\right)}^{2}$
so:
$169 = {x}^{2} + {x}^{2} - 14 x + 49$
$2 {x}^{2} - 14 x - 120 = 0$
${x}_{1 , 2} = \frac{14 \pm \sqrt{196 + 960}}{4} = \frac{14 \pm 34}{4}$
so:
${x}_{1} = 12$
${x}_{2} = - 5$
we use the positive one, giving the length of the longer length as $12$ and shorter: $12 - 7 = 5$

Mar 8, 2016

shorter leg = 5

#### Explanation:

Since this is a right triangle , we can use$\textcolor{b l u e}{\text{ Pythagoras's theorem }}$

If h represents the hypotenuse and a , b the other 2 sides then

${h}^{2} = {a}^{2} + {b}^{2}$

here , let the longer arm = x , so shorter one is then (x-7)

substitute values into formula :

hence ${13}^{2} = {x}^{2} + {\left(x - 7\right)}^{2} = {x}^{2} + {x}^{2} - 14 x + 49$

so $2 {x}^{2} - 14 x + 49 = 169$

This is a quadratic function so equate to zero to solve.

$2 {x}^{2} - 14 x - 120 = 0$

factorising: $2 \left({x}^{2} - 7 x - 60\right) = 0$

To factor ${x}^{2} - 7 x - 60$ require factors of -60 which sum to -7
These are 5 and - 12

$\Rightarrow 2 \left(x - 12\right) \left(x + 5\right) = 0 \Rightarrow x = - 5 , x = 12$

now x > 0 hence x = 12 and so short leg = x-7 = 12 - 7 = 5

Mar 8, 2016

The length of shorter length is $5$

#### Explanation:

Consider the diagram

Use the pythagoras theorem

color(brown)(x^2+y^2=h^2

Where,

$h =$Hypotenuse
$x$ $\mathmr{and}$ $y =$ other two sides

$\therefore {\left(x\right)}^{2} + {\left(x - 7\right)}^{2} = {13}^{2}$

Use the formula color(brown)((a-b)^2=a^2-2ab+b^2#

$\rightarrow {x}^{2} + {x}^{2} - 14 x + 49 = 169$

$\rightarrow 2 {x}^{2} - 14 x + 49 = 169$

$\rightarrow 2 {x}^{2} - 14 x + 49 - 169 = 0$

$\rightarrow 2 {x}^{2} - 14 x - 120 = 0$

Factor it out

$\rightarrow 2 \left(x - 12\right) \left(x + 5\right) = 0$

Remove the $2$

$\rightarrow \left(x - 12\right) \left(x + 5\right) = 0$

If you solve for it you get,

$x = 12 , - 5$

$x > 0 \therefore x = 12$

$x = 12 , x - 7 = 12 - 7 = 5$