Question

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

Answer 1

Length is `12` meters

Explanation:

We can use the Theorem of Pythagoras.

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

By Pythagoras' Theorem:

`x^2 + (2x+2)^2 = 13^2" "larr`square the binomial

`x^2 + 4x^2 + 8x +4 = 169" "larr` make it = 0

`5x^2 +8x +4-169 =0`

`5x^2 +8x -165 = 0`

Find factors of 5 and 165 which subtract to give 8
Note that `165 = 5 xx33`

`33-25 = 8`

`(x-5)(5x +33)=0" "` set each factor = 0

`x-5 = 0 " "rarr x = 5`

`5x+33=0" "rarr 5x = -33` Reject the negative value

If `x-5 " "rarr 2x+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" "and 5:12:13" "and " "7:24:25`

Note that `5 xx2+2 = 12" "larr` this fits what we want.

`5^2 +12^2 = 25+144 = 169`
`13^2 = 169`