Given that $a = 2$ and $b = 2\sqrt{3}$, we can use the Pythagorean theorem to find the length of the hypotenuse $c$ of a right triangle with sides $a$ and $b$.

The Pythagorean theorem states:

$c = \sqrt{a^2 + b^2}$

Substituting the values of $a$ and $b$ into the formula:

$c = \sqrt{2^2 + (2\sqrt{3})^2}$

Now, calculate each term inside the square root:

1. $2^2 = 4$
2. $(2\sqrt{3})^2 = 2^2 \cdot (\sqrt{3})^2 = 4 \cdot 3 = 12$

Add these results together:

$c = \sqrt{4 + 12}$

$c = \sqrt{16}$

Since $\sqrt{16} = 4$, the length of the hypotenuse $c$ is $4$.

Summary

Given $a = 2$ and $b = 2\sqrt{3}$, using the Pythagorean theorem, the hypotenuse $c$ is $4$.

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Related Questions

1. How do you apply the Pythagorean theorem to different types of triangles?
2. What are some techniques to simplify expressions with square roots?
3. Can you find the length of the hypotenuse if both legs of the triangle are irrational numbers?
4. How does the Pythagorean theorem extend to three dimensions?
5. What is the history of the Pythagorean theorem and its discovery?

Tip

When solving problems using the Pythagorean theorem, ensure all sides are in the same units and carefully square the terms to avoid common mistakes.