Answer :
Let's analyze each statement step-by-step.
### Given:
- Side lengths of the triangle: [tex]\( a = 2 \, \text{in} \)[/tex], [tex]\( b = 5 \, \text{in} \)[/tex], [tex]\( c = 4 \, \text{in} \)[/tex].
### Step-by-Step Analysis:
1. Statement 1:
"The triangle is acute because [tex]\( 2^2 + 5^2 > 4^2 \)[/tex]."
- Calculate [tex]\( 2^2 \)[/tex]: [tex]\( 2^2 = 4 \)[/tex].
- Calculate [tex]\( 5^2 \)[/tex]: [tex]\( 5^2 = 25 \)[/tex].
- Calculate [tex]\( 4^2 \)[/tex]: [tex]\( 4^2 = 16 \)[/tex].
- Now, sum [tex]\( 2^2 \)[/tex] and [tex]\( 5^2 \)[/tex]: [tex]\( 2^2 + 5^2 = 4 + 25 = 29 \)[/tex].
- Compare with [tex]\( 4^2 \)[/tex]: [tex]\( 29 > 16 \)[/tex].
This is true, suggesting the triangle might be acute.
2. Statement 2:
"The triangle is acute because [tex]\( 2 + 4 > 5 \)[/tex]."
- Sum [tex]\( 2 \)[/tex] and [tex]\( 4 \)[/tex]: [tex]\( 2 + 4 = 6 \)[/tex].
- Compare with [tex]\( 5 \)[/tex]: [tex]\( 6 > 5 \)[/tex].
This is true, indicating it's a valid triangle according to the triangle inequality theorem, but it doesn't directly prove it's acute.
3. Statement 3:
"The triangle is not acute because [tex]\( 2^2 + 4^2 < 5^2 \)[/tex]."
- Calculate [tex]\( 2^2 \)[/tex]: [tex]\( 2^2 = 4 \)[/tex].
- Calculate [tex]\( 4^2 \)[/tex]: [tex]\( 4^2 = 16 \)[/tex].
- Calculate [tex]\( 5^2 \)[/tex]: [tex]\( 5^2 = 25 \)[/tex].
- Sum [tex]\( 2^2 \)[/tex] and [tex]\( 4^2 \)[/tex]: [tex]\( 2^2 + 4^2 = 4 + 16 = 20 \)[/tex].
- Compare with [tex]\( 5^2 \)[/tex]: [tex]\( 20 < 25 \)[/tex].
This is true, suggesting that the triangle is not acute, because for an acute triangle, the sum of the squares of any two sides should be greater than the square of the third side.
4. Statement 4:
"The triangle is not acute because [tex]\( 2^2 < 4^2 + 5^2 \)[/tex]."
- Calculate [tex]\( 2^2 \)[/tex]: [tex]\( 2^2 = 4 \)[/tex].
- Calculate [tex]\( 4^2 \)[/tex]: [tex]\( 4^2 = 16 \)[/tex].
- Calculate [tex]\( 5^2 \)[/tex]: [tex]\( 5^2 = 25 \)[/tex].
- Sum [tex]\( 4^2 \)[/tex] and [tex]\( 5^2 \)[/tex]: [tex]\( 4^2 + 5^2 = 16 + 25 = 41 \)[/tex].
- Compare with [tex]\( 2^2 \)[/tex]: [tex]\( 4 < 41 \)[/tex].
This is true as well but indicates that [tex]\( 2^2 \)[/tex] (smallest side) is indeed less than the sum of the squares of the other two sides, which is generally true but not enough to classify the type of triangle directly.
### Conclusion:
Examining the correctness:
- Statement 1 is correct and supported by the calculation that [tex]\( 2^2 + 5^2 > 4^2 \)[/tex].
- Statement 2 is correct but an incomplete reasoning for being acute, as it supports it being a valid triangle but not necessarily acute.
- Statement 3 correctly identifies the triangle is not acute because [tex]\( 2^2 + 4^2 < 5^2 \)[/tex].
- Statement 4 expands on the properties of the comparison but is supplementary.
Thus, the best explanation in relation to the given triangle's acuteness is:
"The triangle is not acute because [tex]\( 2^2 + 4^2 < 5^2 \)[/tex]."
### Given:
- Side lengths of the triangle: [tex]\( a = 2 \, \text{in} \)[/tex], [tex]\( b = 5 \, \text{in} \)[/tex], [tex]\( c = 4 \, \text{in} \)[/tex].
### Step-by-Step Analysis:
1. Statement 1:
"The triangle is acute because [tex]\( 2^2 + 5^2 > 4^2 \)[/tex]."
- Calculate [tex]\( 2^2 \)[/tex]: [tex]\( 2^2 = 4 \)[/tex].
- Calculate [tex]\( 5^2 \)[/tex]: [tex]\( 5^2 = 25 \)[/tex].
- Calculate [tex]\( 4^2 \)[/tex]: [tex]\( 4^2 = 16 \)[/tex].
- Now, sum [tex]\( 2^2 \)[/tex] and [tex]\( 5^2 \)[/tex]: [tex]\( 2^2 + 5^2 = 4 + 25 = 29 \)[/tex].
- Compare with [tex]\( 4^2 \)[/tex]: [tex]\( 29 > 16 \)[/tex].
This is true, suggesting the triangle might be acute.
2. Statement 2:
"The triangle is acute because [tex]\( 2 + 4 > 5 \)[/tex]."
- Sum [tex]\( 2 \)[/tex] and [tex]\( 4 \)[/tex]: [tex]\( 2 + 4 = 6 \)[/tex].
- Compare with [tex]\( 5 \)[/tex]: [tex]\( 6 > 5 \)[/tex].
This is true, indicating it's a valid triangle according to the triangle inequality theorem, but it doesn't directly prove it's acute.
3. Statement 3:
"The triangle is not acute because [tex]\( 2^2 + 4^2 < 5^2 \)[/tex]."
- Calculate [tex]\( 2^2 \)[/tex]: [tex]\( 2^2 = 4 \)[/tex].
- Calculate [tex]\( 4^2 \)[/tex]: [tex]\( 4^2 = 16 \)[/tex].
- Calculate [tex]\( 5^2 \)[/tex]: [tex]\( 5^2 = 25 \)[/tex].
- Sum [tex]\( 2^2 \)[/tex] and [tex]\( 4^2 \)[/tex]: [tex]\( 2^2 + 4^2 = 4 + 16 = 20 \)[/tex].
- Compare with [tex]\( 5^2 \)[/tex]: [tex]\( 20 < 25 \)[/tex].
This is true, suggesting that the triangle is not acute, because for an acute triangle, the sum of the squares of any two sides should be greater than the square of the third side.
4. Statement 4:
"The triangle is not acute because [tex]\( 2^2 < 4^2 + 5^2 \)[/tex]."
- Calculate [tex]\( 2^2 \)[/tex]: [tex]\( 2^2 = 4 \)[/tex].
- Calculate [tex]\( 4^2 \)[/tex]: [tex]\( 4^2 = 16 \)[/tex].
- Calculate [tex]\( 5^2 \)[/tex]: [tex]\( 5^2 = 25 \)[/tex].
- Sum [tex]\( 4^2 \)[/tex] and [tex]\( 5^2 \)[/tex]: [tex]\( 4^2 + 5^2 = 16 + 25 = 41 \)[/tex].
- Compare with [tex]\( 2^2 \)[/tex]: [tex]\( 4 < 41 \)[/tex].
This is true as well but indicates that [tex]\( 2^2 \)[/tex] (smallest side) is indeed less than the sum of the squares of the other two sides, which is generally true but not enough to classify the type of triangle directly.
### Conclusion:
Examining the correctness:
- Statement 1 is correct and supported by the calculation that [tex]\( 2^2 + 5^2 > 4^2 \)[/tex].
- Statement 2 is correct but an incomplete reasoning for being acute, as it supports it being a valid triangle but not necessarily acute.
- Statement 3 correctly identifies the triangle is not acute because [tex]\( 2^2 + 4^2 < 5^2 \)[/tex].
- Statement 4 expands on the properties of the comparison but is supplementary.
Thus, the best explanation in relation to the given triangle's acuteness is:
"The triangle is not acute because [tex]\( 2^2 + 4^2 < 5^2 \)[/tex]."
