Let's break down the problem shown in the image.

1. Part (a): Using the Pythagorean Theorem
   The diagram shows a right triangle with sides of lengths $5 \, \text{cm}$ (adjacent side), $13 \, \text{cm}$ (hypotenuse), and an unknown opposite side.

   The Pythagorean Theorem states: $\text{Hypotenuse}^2 = \text{Opposite}^2 + \text{Adjacent}^2$ Plugging in the given values: $13^2 = x^2 + 5^2$ $169 = x^2 + 25$ $x^2 = 169 - 25 = 144$ $x = \sqrt{144} = 12 \, \text{cm}$ Therefore, the length of the third side (opposite) is 12 cm.

2. Part (b): Ratios
   The ratios that need to be found are:

   • Ratio of the length of the opposite side to the hypotenuse relative to angle $A$: $\frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{12}{13} \approx 0.923$

   • Ratio of the length of the opposite side to the length of the hypotenuse relative to angle $C$: The angle $C$ is the right angle, so the ratio here might be misunderstood. If the question intended another ratio, it needs clarification. But for the ratio of the hypotenuse relative to $\angle A$, it's already covered.

Let me know if you'd like any further clarification!

Here are 5 related questions to think about:

1. How would you calculate the sine, cosine, and tangent for angle $A$ using these sides?
2. What are the values of the sine, cosine, and tangent for angle $B$?
3. Can you use the Pythagorean theorem to find the third side in any right triangle?
4. What are the possible uses of trigonometric ratios in real-life applications?
5. How would the calculations change if the lengths of the triangle were doubled?

Tip: Always remember that the sine of an angle in a right triangle is the ratio of the opposite side to the hypotenuse.