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In the triangle shown above, we have a=16.24 cm,b=22.74 cma=16.24 cm,b=22.74 cm and c=15.16 cmc=15.16 cm.

How long is the side xx? **Give your answer with TWO ( 2 ) decimal places!**

Answer: x≈x≈  cm **(TWO DECIMALS!**

To find the length of side \( x \) in the given figure, we can use the Pythagorean Theorem in the two right triangles.

### Step-by-Step Solution:

1. **Identify Triangles:**
   - The figure consists of two right triangles.
   - Let's call the larger triangle with side \( x \), base \( a + b \), and height \( c \).
   - The smaller right triangle has sides \( a \), \( c \), and the hypotenuse (part of \( x \)).

2. **Larger Triangle:**
   - For the larger triangle, we can express \( x \) using the Pythagorean theorem:
     \[
     x^2 = (a + b)^2 + c^2
     \]

3. **Substitute the given values:**
   - \( a = 16.24 \) cm
   - \( b = 22.74 \) cm
   - \( c = 15.16 \) cm

   Plug these values into the formula:

   \[
   x^2 = (16.24 + 22.74)^2 + 15.16^2
   \]

4. **Calculate \( (a + b) \) and then square it:**
   \[
   (16.24 + 22.74) = 38.98
   \]
   \[
   (38.98)^2 = 1519.20
   \]

5. **Square the height \( c \):**
   \[
   (15.16)^2 = 229.82
   \]

6. **Add the two results to find \( x^2 \):**
   \[
   x^2 = 1519.20 + 229.82 = 1749.02
   \]

7. **Find \( x \) by taking the square root:**
   \[
   x = \sqrt{1749.02} \approx 41.82
   \]

### Answer:
\[
x \approx 41.82 \text{ cm}
\]

Would you like more details, or do you have any questions?

### Related Questions:
1. What are the properties of a right triangle that help in solving such problems?
2. How can we determine the height of a triangle if it's not given directly?
3. What are the other possible methods to calculate the length of side \( x \)?
4. Can the Pythagorean theorem be used for non-right triangles? If so, how?
5. How does the triangle inequality theorem apply to this triangle?

### Tip:
Always double-check calculations by verifying each step, especially when using the Pythagorean theorem in multi-part problems.

In the triangle shown above, we have a = 16.24 cm, b = 22.74 cm, and c = 15.16 cm. How long is the side x? Give your answer with TWO (2) decimal places!

This problem involves finding the length of side x in a triangle with sides a = 16.24 cm, b = 22.74 cm, and c = 15.16 cm. The solution applies the Pythagorean Theorem to solve for x. Ideal for students in Grades 8-10.

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