Question

What is the missing side ?

Answer 1

`x=12`

Explanation:

Method 1: Pythagorean Theorem

We can use the Pythagorean Theorem to solve for
`x`. The Pythagorean Theorem states that in a right triangle, the sum of the squares of the legs' side lengths is equal to the length of the hypotenuse squared. Simply put,
`a^(2) +b^(2) =c^(2)`, where
`a` and
`b` are the legs and
`c` is the hypotenuse. In this case, we know that
`a=x`,
`b=9`, and
`c=15`, so we get:

`a^(2) +b^(2) =c^(2)`

`x^(2) +9^(2) =15^(2)` (Substitute
`a=x`,
`b=9`, and
`c=15` into
`a^(2) +b^(2) =c^(2)`)

`x^(2) +81 =225` (Simplify exponents)

`x^(2) +81-81=225-81` (Subtract
`81` from both sides of the equation to isolate
`x`)

`x^(2) =144` (Simplify)

`\sqrt{x^(2)} =√(144)` (Take the square root of both sides)

`x=12,x=-12` (Simplify, remember that each positive number has two square roots: a positive one and a negative one)

In the context of the situation, we know that
`x=-12` is an extraneous solution because a polygon cannot have negative side lengths. Therefore, the final answer is
`x=12`.

Method 2: Pythagorean Triples

Method 1 works, but there's an easier way to find the value of
`x`. If we look at the given lengths of
`9` and
`15`, we can observe that this triangle is a
`3-4-5` right triangle enlarged by a scale factor of
`3`, because
`3*3=9` and
`5*3=15`. The only side length that's missing is the
`4`. Therefore,
`x=4*3=12`. Hope this helps!