Math riddle worksheet with trigonometry problems and a beach-themed joke.

To solve this problem, we need to find the missing side lengths of the right triangles using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be written as:

\[ c^2 = a^2 + b^2 \]

where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides.

Let's solve each triangle one by one:

1. Triangle 1:
- Given: \( a = 12 \), \( b = 9 \)
- Find \( c \):
\[
c^2 = 12^2 + 9^2
\]
\[
c^2 = 144 + 81
\]
\[
c^2 = 225
\]
\[
c = \sqrt{225} = 15
\]
- Answer: \( c = 15 \)

2. Triangle 2:
- Given: \( a = 12 \), \( c = 13 \)
- Find \( b \):
\[
13^2 = 12^2 + b^2
\]
\[
169 = 144 + b^2
\]
\[
b^2 = 169 - 144
\]
\[
b^2 = 25
\]
\[
b = \sqrt{25} = 5
\]
- Answer: \( b = 5 \)

3. Triangle 3:
- Given: \( a = 12 \), \( b = 5 \)
- Find \( c \):
\[
c^2 = 12^2 + 5^2
\]
\[
c^2 = 144 + 25
\]
\[
c^2 = 169
\]
\[
c = \sqrt{169} = 13
\]
- Answer: \( c = 13 \)

4. Triangle 4:
- Given: \( a = 12 \), \( c = 13 \)
- Find \( b \):
\[
13^2 = 12^2 + b^2
\]
\[
169 = 144 + b^2
\]
\[
b^2 = 169 - 144
\]
\[
b^2 = 25
\]
\[
b = \sqrt{25} = 5
\]
- Answer: \( b = 5 \)

5. Triangle 5:
- Given: \( a = 12 \), \( b = 9 \)
- Find \( c \):
\[
c^2 = 12^2 + 9^2
\]
\[
c^2 = 144 + 81
\]
\[
c^2 = 225
\]
\[
c = \sqrt{225} = 15
\]
- Answer: \( c = 15 \)

6. Triangle 6:
- Given: \( a = 12 \), \( c = 13 \)
- Find \( b \):
\[
13^2 = 12^2 + b^2
\]
\[
169 = 144 + b^2
\]
\[
b^2 = 169 - 144
\]
\[
b^2 = 25
\]
\[
b = \sqrt{25} = 5
\]
- Answer: \( b = 5 \)

Now, let's decode the answer to the riddle using the terms of the answer associated with each question:

1. \( 15 \)
2. \( 5 \)
3. \( 13 \)
4. \( 5 \)
5. \( 15 \)
6. \( 5 \)

The answer to the riddle is: "BECAUSE THEY HAVE SINE AND COSINE TO GET A TAN AND DON'T NEED THE SUN!"

Final Answer:
\[ \boxed{\text{BECAUSE THEY HAVE SINE AND COSINE TO GET A TAN AND DON'T NEED THE SUN!}} \]