Question
The triangle shown is a right triangle having sides a, b, and c. Given that the measurements of a = 12 cm, b = 16 cm, and c = 20 cm, which of these statements is true?
A.
The sum of the 2 smaller sides is equal to the third side.
B.
The absolute value of the difference between the squares of the 2 smaller sides is equal to square of the third side.
C.
The sum of the squares of the 2 smaller sides is equal to the third side.
D.
The sum of the squares of the 2 smaller sides is equal to the square of the third side.
Answers
GPT-4o mini
To analyze the statements given regarding the triangle with sides \( a = 12 \, \text{cm} \), \( b = 16 \, \text{cm} \), and \( c = 20 \, \text{cm} \), we can apply the properties of a right triangle.
1. **Statement A**: The sum of the 2 smaller sides is equal to the third side.
- \( a + b = 12 + 16 = 28 \)
- \( c = 20 \)
- This statement is **false**.
2. **Statement B**: The absolute value of the difference between the squares of the 2 smaller sides is equal to the square of the third side.
- \( a^2 - b^2 = 12^2 - 16^2 = 144 - 256 = -112 \) (taking the absolute value, \( | -112 | = 112 \))
- \( c^2 = 20^2 = 400 \)
- This statement is **false**.
3. **Statement C**: The sum of the squares of the 2 smaller sides is equal to the third side.
- \( a^2 + b^2 = 12^2 + 16^2 = 144 + 256 = 400 \)
- \( c = 20 \) (This statement would verify if we considered \(c\) instead of \(c^2\))
- The statement is incorrectly phrased: it should speak of squares rather than just the sides themselves. Nonetheless, this statement is **false** as stated.
4. **Statement D**: The sum of the squares of the 2 smaller sides is equal to the square of the third side.
- We found in the evaluation of Statement C:
\[
a^2 + b^2 = 400 \quad \text{and} \quad c^2 = 400
\]
- Since \( a^2 + b^2 = c^2 \), this statement is **true**.
Thus, the correct statement is:
**D. The sum of the squares of the 2 smaller sides is equal to the square of the third side.**
1. **Statement A**: The sum of the 2 smaller sides is equal to the third side.
- \( a + b = 12 + 16 = 28 \)
- \( c = 20 \)
- This statement is **false**.
2. **Statement B**: The absolute value of the difference between the squares of the 2 smaller sides is equal to the square of the third side.
- \( a^2 - b^2 = 12^2 - 16^2 = 144 - 256 = -112 \) (taking the absolute value, \( | -112 | = 112 \))
- \( c^2 = 20^2 = 400 \)
- This statement is **false**.
3. **Statement C**: The sum of the squares of the 2 smaller sides is equal to the third side.
- \( a^2 + b^2 = 12^2 + 16^2 = 144 + 256 = 400 \)
- \( c = 20 \) (This statement would verify if we considered \(c\) instead of \(c^2\))
- The statement is incorrectly phrased: it should speak of squares rather than just the sides themselves. Nonetheless, this statement is **false** as stated.
4. **Statement D**: The sum of the squares of the 2 smaller sides is equal to the square of the third side.
- We found in the evaluation of Statement C:
\[
a^2 + b^2 = 400 \quad \text{and} \quad c^2 = 400
\]
- Since \( a^2 + b^2 = c^2 \), this statement is **true**.
Thus, the correct statement is:
**D. The sum of the squares of the 2 smaller sides is equal to the square of the third side.**