Question
8.
Determine whether the given lengths can be sides of a right triangle.
Which of the following are true statements.
The lengths 14, 24 and 26 can be sides of a right triangle. The lengths 30, 72, and 78 can be sides of a right triangle.
The lengths 14, 24 and 26 can be sides of a right triangle. The lengths 30, 72, and 78 can not be sides of a right triangle.
The lengths 14, 24 and 26 can not be sides of a right triangle. The lengths 30, 72, and 78 can not be sides of a right triangle.
The lengths 14, 24 and 26 can not be sides of a right triangle. The lengths 30, 72, and 78 can be sides of a right triangle.
Determine whether the given lengths can be sides of a right triangle.
Which of the following are true statements.
The lengths 14, 24 and 26 can be sides of a right triangle. The lengths 30, 72, and 78 can be sides of a right triangle.
The lengths 14, 24 and 26 can be sides of a right triangle. The lengths 30, 72, and 78 can not be sides of a right triangle.
The lengths 14, 24 and 26 can not be sides of a right triangle. The lengths 30, 72, and 78 can not be sides of a right triangle.
The lengths 14, 24 and 26 can not be sides of a right triangle. The lengths 30, 72, and 78 can be sides of a right triangle.
Answers
GPT-4o mini
To determine whether the given lengths can be sides of a right triangle, we can use the Pythagorean theorem, which states that for a triangle with sides \(a\), \(b\), and hypotenuse \(c\), the following equation must be true:
\[
a^2 + b^2 = c^2
\]
where \(c\) is the longest side.
Let's analyze each set of lengths provided.
1. **For lengths 14, 24, and 26:**
- The longest side is 26. We check if:
\[
14^2 + 24^2 = 26^2
\]
- Calculate \(14^2\), \(24^2\), and \(26^2\):
- \(14^2 = 196\)
- \(24^2 = 576\)
- \(26^2 = 676\)
- Now, check if:
\[
196 + 576 = 676
\]
- \(772 \neq 676\). Thus, lengths 14, 24, and 26 **cannot** form a right triangle.
2. **For lengths 30, 72, and 78:**
- The longest side is 78. We check if:
\[
30^2 + 72^2 = 78^2
\]
- Calculate \(30^2\), \(72^2\), and \(78^2\):
- \(30^2 = 900\)
- \(72^2 = 5184\)
- \(78^2 = 6084\)
- Now check if:
\[
900 + 5184 = 6084
\]
- \(6084 = 6084\). Thus, lengths 30, 72, and 78 **can** form a right triangle.
Now we summarize the findings:
- The lengths **14, 24, and 26 cannot be sides of a right triangle.**
- The lengths **30, 72, and 78 can be sides of a right triangle.**
Thus, the correct statement is:
**The lengths 14, 24 and 26 can not be sides of a right triangle. The lengths 30, 72, and 78 can be sides of a right triangle.**
\[
a^2 + b^2 = c^2
\]
where \(c\) is the longest side.
Let's analyze each set of lengths provided.
1. **For lengths 14, 24, and 26:**
- The longest side is 26. We check if:
\[
14^2 + 24^2 = 26^2
\]
- Calculate \(14^2\), \(24^2\), and \(26^2\):
- \(14^2 = 196\)
- \(24^2 = 576\)
- \(26^2 = 676\)
- Now, check if:
\[
196 + 576 = 676
\]
- \(772 \neq 676\). Thus, lengths 14, 24, and 26 **cannot** form a right triangle.
2. **For lengths 30, 72, and 78:**
- The longest side is 78. We check if:
\[
30^2 + 72^2 = 78^2
\]
- Calculate \(30^2\), \(72^2\), and \(78^2\):
- \(30^2 = 900\)
- \(72^2 = 5184\)
- \(78^2 = 6084\)
- Now check if:
\[
900 + 5184 = 6084
\]
- \(6084 = 6084\). Thus, lengths 30, 72, and 78 **can** form a right triangle.
Now we summarize the findings:
- The lengths **14, 24, and 26 cannot be sides of a right triangle.**
- The lengths **30, 72, and 78 can be sides of a right triangle.**
Thus, the correct statement is:
**The lengths 14, 24 and 26 can not be sides of a right triangle. The lengths 30, 72, and 78 can be sides of a right triangle.**