To determine if three lengths can form a right triangle, we can use the Pythagorean theorem, which states that for a right triangle with legs \( a \) and \( b \), and hypotenuse \( c \), the following must hold:
\[ a^2 + b^2 = c^2 \]
In this case, Isabella has stick lengths of 10 cm, 11 cm, and 13 cm. We will assume the longest stick (13 cm) is the hypotenuse and check if the other two lengths satisfy the Pythagorean theorem.
Let's assign the lengths as follows:
- \( a = 10 , \text{cm} \)
- \( b = 11 , \text{cm} \)
- \( c = 13 , \text{cm} \)
Now, we will check if \( a^2 + b^2 = c^2 \):
Calculating \( a^2 \) and \( b^2 \): \[ a^2 = 10^2 = 100 \] \[ b^2 = 11^2 = 121 \]
Calculating \( c^2 \): \[ c^2 = 13^2 = 169 \]
Now we add \( a^2 \) and \( b^2 \): \[ a^2 + b^2 = 100 + 121 = 221 \]
Now we compare \( a^2 + b^2 \) with \( c^2 \): \[ 221 \neq 169 \]
Since \( a^2 + b^2 \) does not equal \( c^2 \), the sticks of lengths 10 cm, 11 cm, and 13 cm cannot form a right triangle.
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