Question

what is the value of x in the diagram? what is the value of x in the diagram?

Explanation:

Step1: Analyze the right - triangle

We have a right - triangle with an angle of 30° and an adjacent side of length 9. We know that in a right - triangle, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Here, $\theta = 30^{\circ}$ and the adjacent side to the 30° angle is 9. Let the hypotenuse of the smaller right - triangle be $y$. So, $\cos30^{\circ}=\frac{9}{y}$. Since $\cos30^{\circ}=\frac{\sqrt{3}}{2}$, we have $\frac{\sqrt{3}}{2}=\frac{9}{y}$, and $y = \frac{18}{\sqrt{3}}$.

Step2: Consider the larger right - triangle

The larger right - triangle has an angle of 45°. In a 45 - 45-90 right - triangle, the ratio of the sides is $1:1:\sqrt{2}$. The side of the 45 - 45-90 right - triangle is the hypotenuse of the smaller 30 - 60 - 90 right - triangle, which is $y=\frac{18}{\sqrt{3}}$. For a 45 - 45-90 right - triangle, if one of the legs is $a$ and the hypotenuse is $x$, then $x = a\sqrt{2}$. Here $a=\frac{18}{\sqrt{3}}$, so $x=\frac{18\sqrt{2}}{\sqrt{3}}$.

Answer:

$\frac{18\sqrt{2}}{\sqrt{3}}$