Answer:
[tex]x=12[/tex]
Step-by-step explanation:
Method 1: Pythagorean Theorem
We can use the Pythagorean Theorem to solve for [tex]x[/tex]. 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, [tex]a^{2} +b^{2} =c^{2}[/tex], where [tex]a[/tex] and [tex]b[/tex] are the legs and [tex]c[/tex] is the hypotenuse. In this case, we know that [tex]a=x[/tex], [tex]b=9[/tex], and [tex]c=15[/tex], so we get:
[tex]a^{2} +b^{2} =c^{2}[/tex]
[tex]x^{2} +9^{2} =15^{2}[/tex] (Substitute [tex]a=x[/tex], [tex]b=9[/tex], and [tex]c=15[/tex] into [tex]a^{2} +b^{2} =c^{2}[/tex])
[tex]x^{2} +81 =225[/tex] (Simplify exponents)
[tex]x^{2} +81-81=225-81[/tex] (Subtract [tex]81[/tex] from both sides of the equation to isolate [tex]x[/tex])
[tex]x^{2} =144[/tex] (Simplify)
[tex]\sqrt{x^{2}} =\sqrt{144}[/tex] (Take the square root of both sides)
[tex]x=12,x=-12[/tex] (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 [tex]x=-12[/tex] is an extraneous solution because a polygon cannot have negative side lengths. Therefore, the final answer is [tex]x=12[/tex].
Method 2: Pythagorean Triples
Method 1 works, but there's an easier way to find the value of [tex]x[/tex]. If we look at the given lengths of [tex]9[/tex] and [tex]15[/tex], we can observe that this triangle is a [tex]3-4-5[/tex] right triangle enlarged by a scale factor of [tex]3[/tex], because [tex]3*3=9[/tex] and [tex]5*3=15[/tex]. The only side length that's missing is the [tex]4[/tex]. Therefore, [tex]x=4*3=12[/tex]. Hope this helps!
