This worksheet covers the Pythagorean Theorem and its converse, applying it to find missing side lengths in right triangles. Let's solve each question step-by-step.

Example 1: Find the value of $x$

The Pythagorean Theorem states: $a^2 + b^2 = c^2$ where $c$ is the hypotenuse (the side opposite the right angle), and $a$ and $b$ are the other two sides.

1. Problem 1: $a = 3$, $b = 4$, $c = x$

   • $3^2 + 4^2 = x^2$
   • $9 + 16 = x^2$
   • $x^2 = 25$
   • $x = 5$

2. Problem 2: $a = 9$, $b = x$, $c = 15$

   • $9^2 + x^2 = 15^2$
   • $81 + x^2 = 225$
   • $x^2 = 144$
   • $x = 12$

3. Problem 3: $a = x$, $b = 25$, $c = 65$

   • $x^2 + 25^2 = 65^2$
   • $x^2 + 625 = 4225$
   • $x^2 = 3600$
   • $x = 60$

4. Problem 4: $a = x$, $b = 9$, $c = 41$

   • $x^2 + 9^2 = 41^2$
   • $x^2 + 81 = 1681$
   • $x^2 = 1600$
   • $x = 40$

5. Problem 5: $a = 16$, $b = x$, $c = 33$

   • $16^2 + x^2 = 33^2$
   • $256 + x^2 = 1089$
   • $x^2 = 833$
   • $x \approx 28.9$

6. Problem 6: $a = 11$, $b = x$, $c = 28$

   • $11^2 + x^2 = 28^2$
   • $121 + x^2 = 784$
   • $x^2 = 663$
   • $x \approx 25.7$

Example 2: Use a Pythagorean Triple to find the value of $x$

Pythagorean triples are sets of integers that satisfy $a^2 + b^2 = c^2$, such as $(3, 4, 5)$ and $(5, 12, 13)$.

7. Problem 7: $8, 15, 17$ triple

   • $x = 15$

8. Problem 8: $9, 12, 15$ triple scaled by 5: $(15, 20, 25)$

   • $x = 25$

9. Problem 9: $x, 96, 28$

   • Use Pythagorean Theorem:
   • $x^2 + 28^2 = 96^2$

10. **Problem 10