To determine how many squares will fill the large square in relation to the Pythagorean theorem, we can use the theorem itself. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse \( c \) is equal to the sum of the squares of the lengths of the other two sides (legs), \( a \) and \( b \):
\[ c^2 = a^2 + b^2 \]
If we assume that Derek has given measurements related to the legs \( a \) and \( b \) of the triangle, we can follow these steps:
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Identify the squares: If Derek started counting with 10, then we need to understand what that means. If he numbered the squares starting from 10, it's possible that:
- The lengths of the two legs correspond with the squares he numbered.
- For example, if one leg has length \( a \) and the other has length \( b \), the number of unit squares that fit on each leg will correspond to their lengths.
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Find \( a \) and \( b \): Assuming that the numbers counted correspond to specific leg lengths:
- If one leg had 10 squares starting from 10, it could suggest that the total number of squares on that leg is \( n \) where \( n \) includes 10 as one of the counts.
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Calculate the total area: Since the large square is formed by the right triangle, filling the large square involves calculating \( c^2 \), where:
- \( c = \sqrt{a^2 + b^2} \)
- Thus, to fill the large square corresponding to the right triangle formed by sides \( a \) and \( b \), you would ultimately be looking at \( a^2 + b^2 \).
To give a precise answer, we need the specific lengths of \( a \) and \( b \) from Derek's squares. If you can provide the lengths or more context, I can help calculate the exact number of squares that will fill the large square. Otherwise, you can use the steps above to find the answer based on the provided squares.