Question

What is the missing side ?

Answer 1

Answer:

$x=12$

Step-by-step 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)

$\sqrt{x^{2}} =\sqrt{144}$ (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!