We have a triangle with one side of length 9 cm and an angle of 55 degrees opposite side $x$. Side $y$ is adjacent to the 55-degree angle and is the hypotenuse of a right triangle where side $x$ is opposite the right angle. We are asked to determine the lengths of sides $x$ and $y$.

1. Problem Description

We have a triangle with one side of length 9 cm and an angle of 55 degrees opposite side $x$. Side $y$ is adjacent to the 55-degree angle and is the hypotenuse of a right triangle where side $x$ is opposite the right angle. We are asked to determine the lengths of sides $x$ and $y$.

2. Solution Steps

Since the triangle with sides $x$ and $y$ is a right triangle, we can use trigonometric functions to relate the sides and the given angle.

We have the side opposite the angle and the side adjacent to the angle. Thus we can find y using the cosine rule in the larger triangle. Let the third angle be A. Then,

$A + 55 + 90 = 180$, therefore $A = 35$ degrees.

Since we know the length of the side opposite to the 55 degree angle and we know that is 9, we can set up the following relationships:

$\frac{x}{\sin(55)} = \frac{9}{\sin(90)}$

$\frac{y}{\sin(35)} = \frac{9}{\sin(90)}$

Since $\sin(90) = 1$, we have

$x = 9 \sin(55)$

$y = 9 \sin(35)$

$x = 9 \sin(55^\circ)$

$x \approx 9 \times 0.81915 \approx 7.372$

$y = 9 \sin(35^\circ)$

$y \approx 9 \times 0.57358 \approx 5.162$

3. Final Answer

$x = 9 \sin(55^\circ)$

$y = 9 \sin(35^\circ)$