Question
If a triangle has sides measuring 60 in., 75 in., and 45 in. in length, is it a right triangle?
Answers
GPT-4o mini
To determine if a triangle with sides measuring 60 inches, 75 inches, and 45 inches is a right triangle, we can use the Pythagorean theorem. The theorem states that for a triangle with sides \(a\), \(b\), and \(c\) (where \(c\) is the longest side), the triangle is a right triangle if:
\[
a^2 + b^2 = c^2
\]
First, identify the longest side. In this case, the sides are 60 in., 75 in., and 45 in. The longest side is 75 in.
Now, we will assign the sides as:
- \(a = 60\)
- \(b = 45\)
- \(c = 75\)
Next, we calculate \(a^2\), \(b^2\), and \(c^2\):
\[
a^2 = 60^2 = 3600
\]
\[
b^2 = 45^2 = 2025
\]
\[
c^2 = 75^2 = 5625
\]
Now add \(a^2\) and \(b^2\):
\[
a^2 + b^2 = 3600 + 2025 = 5625
\]
Now compare this sum to \(c^2\):
\[
c^2 = 5625
\]
Since \(a^2 + b^2\) equals \(c^2\):
\[
3600 + 2025 = 5625
\]
This means that \(a^2 + b^2 = c^2\) is true.
Therefore, the triangle with sides measuring 60 inches, 75 inches, and 45 inches is indeed a right triangle.
\[
a^2 + b^2 = c^2
\]
First, identify the longest side. In this case, the sides are 60 in., 75 in., and 45 in. The longest side is 75 in.
Now, we will assign the sides as:
- \(a = 60\)
- \(b = 45\)
- \(c = 75\)
Next, we calculate \(a^2\), \(b^2\), and \(c^2\):
\[
a^2 = 60^2 = 3600
\]
\[
b^2 = 45^2 = 2025
\]
\[
c^2 = 75^2 = 5625
\]
Now add \(a^2\) and \(b^2\):
\[
a^2 + b^2 = 3600 + 2025 = 5625
\]
Now compare this sum to \(c^2\):
\[
c^2 = 5625
\]
Since \(a^2 + b^2\) equals \(c^2\):
\[
3600 + 2025 = 5625
\]
This means that \(a^2 + b^2 = c^2\) is true.
Therefore, the triangle with sides measuring 60 inches, 75 inches, and 45 inches is indeed a right triangle.