Question
Use the converse of the Pythagorean theorem to determine whether a right triangle
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GPT-4o mini
To determine whether a triangle is a right triangle, you can use the converse of the Pythagorean theorem. The 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. The converse states the following:
If \( c \) is the length of the longest side (hypotenuse) and \( a \) and \( b \) are the lengths of the other two sides, then the triangle is a right triangle if:
\[
c^2 = a^2 + b^2
\]
### Steps to Use the Converse of the Pythagorean Theorem:
1. **Identify the lengths** of all three sides of the triangle and label them accordingly as \( a \), \( b \), and \( c \), ensuring that \( c \) is the longest side.
2. **Calculate \( a^2 + b^2 \)**.
3. **Calculate \( c^2 \)**.
4. **Compare the results**:
- If \( c^2 = a^2 + b^2 \), then the triangle is a right triangle.
- If \( c^2 \neq a^2 + b^2 \), then the triangle is not a right triangle.
### Example
Suppose you have a triangle with sides of lengths 3, 4, and 5.
1. Identify the sides:
- Let \( a = 3 \), \( b = 4 \), and \( c = 5 \) (since 5 is the longest side).
2. Calculate \( a^2 + b^2 \):
\[
a^2 + b^2 = 3^2 + 4^2 = 9 + 16 = 25
\]
3. Calculate \( c^2 \):
\[
c^2 = 5^2 = 25
\]
4. Compare the results:
\[
c^2 = a^2 + b^2 \quad \Rightarrow \quad 25 = 25
\]
Since \( c^2 = a^2 + b^2 \) is true, the triangle with sides 3, 4, and 5 is a right triangle.
You can use these steps with any triangle to determine if it is a right triangle.
If \( c \) is the length of the longest side (hypotenuse) and \( a \) and \( b \) are the lengths of the other two sides, then the triangle is a right triangle if:
\[
c^2 = a^2 + b^2
\]
### Steps to Use the Converse of the Pythagorean Theorem:
1. **Identify the lengths** of all three sides of the triangle and label them accordingly as \( a \), \( b \), and \( c \), ensuring that \( c \) is the longest side.
2. **Calculate \( a^2 + b^2 \)**.
3. **Calculate \( c^2 \)**.
4. **Compare the results**:
- If \( c^2 = a^2 + b^2 \), then the triangle is a right triangle.
- If \( c^2 \neq a^2 + b^2 \), then the triangle is not a right triangle.
### Example
Suppose you have a triangle with sides of lengths 3, 4, and 5.
1. Identify the sides:
- Let \( a = 3 \), \( b = 4 \), and \( c = 5 \) (since 5 is the longest side).
2. Calculate \( a^2 + b^2 \):
\[
a^2 + b^2 = 3^2 + 4^2 = 9 + 16 = 25
\]
3. Calculate \( c^2 \):
\[
c^2 = 5^2 = 25
\]
4. Compare the results:
\[
c^2 = a^2 + b^2 \quad \Rightarrow \quad 25 = 25
\]
Since \( c^2 = a^2 + b^2 \) is true, the triangle with sides 3, 4, and 5 is a right triangle.
You can use these steps with any triangle to determine if it is a right triangle.