Question

solve for x. leave your answer in simplest radical form.

Explanation:

Step1: Find the height of the left triangle

Let the height of the left right - triangle (with base 2 and hypotenuse 4) be \( h \). Using the Pythagorean theorem \( a^{2}+b^{2}=c^{2} \), where \( c = 4 \) and \( a=2 \), we have \( 2^{2}+h^{2}=4^{2} \).
So \( 4 + h^{2}=16 \), then \( h^{2}=16 - 4=12 \), and \( h=\sqrt{12} = 2\sqrt{3} \).

Step2: Solve for \( x \) in the right - triangle on the right

In the right - triangle on the right, the two legs are \( h = 2\sqrt{3} \) and 3, and the hypotenuse is \( x \). Using the Pythagorean theorem \( x^{2}=(2\sqrt{3})^{2}+3^{2} \).
First, calculate \( (2\sqrt{3})^{2}=2^{2}\times(\sqrt{3})^{2}=4\times3 = 12 \), and \( 3^{2}=9 \). Then \( x^{2}=12 + 9=21 \), so \( x=\sqrt{21} \).

Answer:

\(\sqrt{21}\)