# If a=5 and c=13, how do you find b?

Oct 17, 2015

#### Answer:

Use Pythagoras and rearrange to find $b = 12$

#### Explanation:

Assuming we're dealing with a right angled triangle with legs of lengths $a$, $b$ and hypotenuse of length $c$, Pythagoras theorem tells us:

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

Subtracting ${a}^{2}$ from both sides, we get:

${b}^{2} = {c}^{2} - {a}^{2}$

Then taking the square root of both sides, we get:

$b = \sqrt{{c}^{2} - {a}^{2}}$

We are told that $a = 5$ and $c = 13$, so

$b = \sqrt{{13}^{2} - {5}^{2}} = \sqrt{169 - 25} = \sqrt{144} = 12$

Bonus

The $5 , 12 , 13$ triangle is the second one in a sequence of right angled triangles that starts with the $3 , 4 , 5$ triangle.

$a = 2 k + 3$

$b = \frac{{a}^{2} - 1}{2} = 2 {k}^{2} + 6 k + 4$

$c = \frac{{a}^{2} + 1}{2} = 2 {k}^{2} + 6 k + 5$

This gives us right angled triangles with sides:

$k = 0 :$ $3 , 4 , 5$

$k = 1 :$ $5 , 12 , 13$

$k = 2 :$ $7 , 24 , 25$

$k = 3 :$ $9 , 40 , 41$

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