Question
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 cannot 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 be sides of a right triangle.
The lengths 14, 24 and 26 cannot be sides of a right triangle. The lengths 30, 72, and 78 cannot be sides of a right triangle.
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.
The lengths 14, 24 and 26 can be sides of a right triangle. The lengths 30, 72, and 78 cannot 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 be sides of a right triangle.
The lengths 14, 24 and 26 cannot be sides of a right triangle. The lengths 30, 72, and 78 cannot be sides of a right triangle.
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.
Answers
GPT-4o mini
To determine whether the given lengths can be sides of a right triangle, we can use the Pythagorean theorem. According to this theorem, for three lengths a, b, and c (where c is the longest side), the lengths can form a right triangle if:
\[ a^2 + b^2 = c^2 \]
Let's evaluate each set of lengths.
1. **Lengths 14, 24, and 26:**
- Here, the longest side is 26 (c).
- We check if \( 14^2 + 24^2 = 26^2 \):
\[
14^2 = 196
\]
\[
24^2 = 576
\]
\[
26^2 = 676
\]
\[
196 + 576 = 772 \quad \text{(not equal to 676)}
\]
So, **14, 24, and 26 cannot form a right triangle.**
2. **Lengths 30, 72, and 78:**
- Here, the longest side is 78 (c).
- We check if \( 30^2 + 72^2 = 78^2 \):
\[
30^2 = 900
\]
\[
72^2 = 5184
\]
\[
78^2 = 6084
\]
\[
900 + 5184 = 6084 \quad \text{(equal to 6084)}
\]
So, **30, 72, and 78 can form a right triangle.**
Now let's 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**.
Now, matching these results with the provided statements:
1. **The lengths 14, 24 and 26 can be sides of a right triangle. The lengths 30, 72, and 78 cannot be sides of a right triangle.** (False)
2. **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.** (False)
3. **The lengths 14, 24 and 26 cannot be sides of a right triangle. The lengths 30, 72, and 78 cannot be sides of a right triangle.** (False)
4. **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.** (True)
Therefore, the correct statement is:
**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.**
\[ a^2 + b^2 = c^2 \]
Let's evaluate each set of lengths.
1. **Lengths 14, 24, and 26:**
- Here, the longest side is 26 (c).
- We check if \( 14^2 + 24^2 = 26^2 \):
\[
14^2 = 196
\]
\[
24^2 = 576
\]
\[
26^2 = 676
\]
\[
196 + 576 = 772 \quad \text{(not equal to 676)}
\]
So, **14, 24, and 26 cannot form a right triangle.**
2. **Lengths 30, 72, and 78:**
- Here, the longest side is 78 (c).
- We check if \( 30^2 + 72^2 = 78^2 \):
\[
30^2 = 900
\]
\[
72^2 = 5184
\]
\[
78^2 = 6084
\]
\[
900 + 5184 = 6084 \quad \text{(equal to 6084)}
\]
So, **30, 72, and 78 can form a right triangle.**
Now let's 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**.
Now, matching these results with the provided statements:
1. **The lengths 14, 24 and 26 can be sides of a right triangle. The lengths 30, 72, and 78 cannot be sides of a right triangle.** (False)
2. **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.** (False)
3. **The lengths 14, 24 and 26 cannot be sides of a right triangle. The lengths 30, 72, and 78 cannot be sides of a right triangle.** (False)
4. **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.** (True)
Therefore, the correct statement is:
**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.**