Question
Paul is building a bridge for a school project and wants to make a right triangle out of three popsicle sticks that measure 8 cm, 10 cm and 18 cm. Can he make a right triangle out of these lengths?
Answers
GPT-4o mini
To determine if Paul can form a right triangle with popsicle sticks measuring 8 cm, 10 cm, and 18 cm, we can apply the Pythagorean theorem. This theorem states that in a right triangle with sides \( a \), \( b \), and hypotenuse \( c \), the following relationship holds:
\[
a^2 + b^2 = c^2
\]
Firstly, we assume the longest side (the hypotenuse \( c \)) is 18 cm, and the other two sides \( a \) and \( b \) are 8 cm and 10 cm.
Let's calculate \( a^2 + b^2 \):
\[
8^2 + 10^2 = 64 + 100 = 164
\]
Now we calculate \( c^2 \):
\[
c^2 = 18^2 = 324
\]
Next, we check if \( a^2 + b^2 = c^2 \):
\[
164 \neq 324
\]
Since \( a^2 + b^2 \) does not equal \( c^2 \), the lengths 8 cm, 10 cm, and 18 cm cannot form a right triangle.
Thus, the answer is **No, Paul cannot make a right triangle out of these lengths.**
\[
a^2 + b^2 = c^2
\]
Firstly, we assume the longest side (the hypotenuse \( c \)) is 18 cm, and the other two sides \( a \) and \( b \) are 8 cm and 10 cm.
Let's calculate \( a^2 + b^2 \):
\[
8^2 + 10^2 = 64 + 100 = 164
\]
Now we calculate \( c^2 \):
\[
c^2 = 18^2 = 324
\]
Next, we check if \( a^2 + b^2 = c^2 \):
\[
164 \neq 324
\]
Since \( a^2 + b^2 \) does not equal \( c^2 \), the lengths 8 cm, 10 cm, and 18 cm cannot form a right triangle.
Thus, the answer is **No, Paul cannot make a right triangle out of these lengths.**