The triangle in the image is a right triangle, and the measure of $c$ can be determined using the Pythagorean Theorem:

$c^2 = a^2 + b^2$

Here:

• $a = 8$
• $b = 12$

Step 1: Apply the formula

$c^2 = 8^2 + 12^2$

$c^2 = 64 + 144$

$c^2 = 208$

Step 2: Solve for $c$

$c = \sqrt{208}$

Step 3: Simplify the square root

The prime factorization of 208 is: $208 = 16 \times 13 = (4^2) \times 13$

$\sqrt{208} = \sqrt{(4^2) \times 13} = 4 \sqrt{13}$

Thus: $c = 4\sqrt{13}$

Final Answer:

The measure of $c$ is $4\sqrt{13}$.

Let me know if you'd like further explanation! 😊

Relative Questions:

1. What is the Pythagorean Theorem, and how is it applied in right triangles?
2. How do you simplify square roots with prime factorization?
3. Why is $\sqrt{208} = 4\sqrt{13}$ and not left unsimplified?
4. What are some real-life applications of the Pythagorean Theorem?
5. How can you identify when to use the Pythagorean Theorem in geometry problems?

Tip:

Always check if square roots can be simplified by finding the largest perfect square factor! This helps in presenting the most simplified answer.