find the value of x in the triangle shown below.

Explanation:

Step1: Identify the theorem

Use the Pythagorean theorem $a^{2}+b^{2}=c^{2}$ for right - triangles. Here $a = 1$, $b = 4$, and $c=x$.

Step2: Substitute values

Substitute $a = 1$ and $b = 4$ into the Pythagorean theorem: $1^{2}+4^{2}=x^{2}$.

Step3: Calculate the left - hand side

$1^{2}=1$, $4^{2}=16$, so $1 + 16=x^{2}$, which gives $x^{2}=17$.

Step4: Solve for x

Take the square root of both sides. Since $x$ represents the length of a side of a triangle, we take the positive square root. So $x=\sqrt{17}$.

Answer:

$x = \sqrt{17}$