https://file.aiquickdraw.com/mathbot/file/1727178728464wmrn22d1.jpg

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!

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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.

Use the following diagram to answer the questions. (a) Use the Pythagorean Theorem to find the length of the third side. (b) Find the ratio of the length of the opposite side to the length of the hypotenuse relative to angle A.

Learn how to solve for the missing side of a right triangle using the Pythagorean Theorem. Also, calculate trigonometric ratios, specifically the ratio of the opposite side to the hypotenuse relative to a given angle. Suitable for students in Grades 8-10.

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