Question

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

Explanation:

Step1: Find the height (let's call it \( h \)) of the left right - triangle

In the left right - triangle with hypotenuse \( 4 \) and base \( 2 \), we use the Pythagorean theorem \( a^{2}+b^{2}=c^{2} \), where \( c = 4 \), \( a = 2 \), and we want to find \( b=h \).
So, \( h^{2}+2^{2}=4^{2} \)
\( h^{2}+4 = 16 \)
\( h^{2}=16 - 4=12 \)
\( h=\sqrt{12}=2\sqrt{3} \) (we take the positive value since height is a length)

Step2: Use the height to find \( x \) in the right - triangle on the right

In the right - triangle on the right, we know one leg is \( h = 2\sqrt{3} \) and the other leg is \( 3 \), and we want to find the hypotenuse \( x \).
Using the Pythagorean theorem \( x^{2}=h^{2}+3^{2} \)
Substitute \( h^{2}=12 \) into the formula: \( x^{2}=12 + 9=21 \)
Then \( x=\sqrt{21} \) (we take the positive value since \( x \) represents a length)

Answer:

\( \sqrt{21} \)