The diagonal of a rectangle is 13 meters. The length is 2 meters more than twice the width. What is the length?

Dec 21, 2016

Length is $12$ meters

Explanation:

We can use the Theorem of Pythagoras.

Let the width be $x$
The length is then $2 x + 2$

By Pythagoras' Theorem:

${x}^{2} + {\left(2 x + 2\right)}^{2} = {13}^{2} \text{ } \leftarrow$square the binomial

${x}^{2} + 4 {x}^{2} + 8 x + 4 = 169 \text{ } \leftarrow$ make it = 0

$5 {x}^{2} + 8 x + 4 - 169 = 0$

$5 {x}^{2} + 8 x - 165 = 0$

Find factors of 5 and 165 which subtract to give 8
Note that $165 = 5 \times 33$

$33 - 25 = 8$

$\left(x - 5\right) \left(5 x + 33\right) = 0 \text{ }$ set each factor = 0

$x - 5 = 0 \text{ } \rightarrow x = 5$

$5 x + 33 = 0 \text{ } \rightarrow 5 x = - 33$ Reject the negative value

If $x - 5 \text{ } \rightarrow 2 x + 2 = 12$

We could also have guessed at this outcome using the
Pythagorean triples... 13 is a clue!

The common triples are:

$3 : 4 : 5 \text{ "and 5:12:13" "and " } 7 : 24 : 25$

Note that $5 \times 2 + 2 = 12 \text{ } \leftarrow$ this fits what we want.

${5}^{2} + {12}^{2} = 25 + 144 = 169$
${13}^{2} = 169$