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Sagot :
To find the height of an equilateral triangle with a side length of [tex]\( s \)[/tex] units, we use a specific formula derived from geometric principles. For an equilateral triangle, all sides are equal, and the altitude splits the triangle into two 30-60-90 right triangles.
### Step-by-step Calculation:
1. Formula for the height of an equilateral triangle:
The height [tex]\( h \)[/tex] of an equilateral triangle with side length [tex]\( s \)[/tex] can be determined by the formula:
[tex]\[ h = \frac{\sqrt{3}}{2} s \][/tex]
2. Evaluate the given options:
Let's compare each provided option with the correct height formula.
- [tex]\(\frac{s}{2} \sqrt{2}\)[/tex]:
[tex]\[ \frac{s}{2} \sqrt{2} \quad (\approx 0.707 \text{ for } s = 1) \][/tex]
- [tex]\(\frac{5}{2} \sqrt{3}\)[/tex]:
[tex]\[ \frac{5}{2} \sqrt{3} \quad (\approx 4.33 \text{ for } s = 1) \][/tex]
- [tex]\( s \sqrt{2} \)[/tex]:
[tex]\[ s \sqrt{2} \quad (\approx 1.414 \text{ for } s = 1) \][/tex]
- [tex]\( 5 \sqrt{3} \)[/tex]:
[tex]\[ 5 \sqrt{3} \quad (\approx 8.66 \text{ for } s = 1) \][/tex]
3. Comparison with the correct height:
Based on the derivations:
- The correct height derived from [tex]\(\frac{\sqrt{3}}{2} s\)[/tex] gives approximately [tex]\(0.866 \, s\)[/tex] when [tex]\( s = 1 \)[/tex].
Comparing this with the evaluated options:
[tex]\[ 0.866 \approx \frac{\sqrt{3}}{2} \quad \text{(actual height)} \][/tex]
[tex]\[ 0.866 \neq 0.707 \, (option 1) \][/tex]
[tex]\[ 0.866 \neq 4.33 \, (option 2) \][/tex]
[tex]\[ 0.866 \neq 1.414 \, (option 3) \][/tex]
[tex]\[ 0.866 \neq 8.66 \, (option 4) \][/tex]
Since none of the provided options match the calculated height [tex]\(\frac{\sqrt{3}}{2} s\)[/tex], it suggests that there is either an error in the given choices, or the true answer is not provided among the options.
After careful evaluation and comparison, none of the given options accurately represent the height of the triangular base of the pyramid based on the correct geometric formula for an equilateral triangle.
### Step-by-step Calculation:
1. Formula for the height of an equilateral triangle:
The height [tex]\( h \)[/tex] of an equilateral triangle with side length [tex]\( s \)[/tex] can be determined by the formula:
[tex]\[ h = \frac{\sqrt{3}}{2} s \][/tex]
2. Evaluate the given options:
Let's compare each provided option with the correct height formula.
- [tex]\(\frac{s}{2} \sqrt{2}\)[/tex]:
[tex]\[ \frac{s}{2} \sqrt{2} \quad (\approx 0.707 \text{ for } s = 1) \][/tex]
- [tex]\(\frac{5}{2} \sqrt{3}\)[/tex]:
[tex]\[ \frac{5}{2} \sqrt{3} \quad (\approx 4.33 \text{ for } s = 1) \][/tex]
- [tex]\( s \sqrt{2} \)[/tex]:
[tex]\[ s \sqrt{2} \quad (\approx 1.414 \text{ for } s = 1) \][/tex]
- [tex]\( 5 \sqrt{3} \)[/tex]:
[tex]\[ 5 \sqrt{3} \quad (\approx 8.66 \text{ for } s = 1) \][/tex]
3. Comparison with the correct height:
Based on the derivations:
- The correct height derived from [tex]\(\frac{\sqrt{3}}{2} s\)[/tex] gives approximately [tex]\(0.866 \, s\)[/tex] when [tex]\( s = 1 \)[/tex].
Comparing this with the evaluated options:
[tex]\[ 0.866 \approx \frac{\sqrt{3}}{2} \quad \text{(actual height)} \][/tex]
[tex]\[ 0.866 \neq 0.707 \, (option 1) \][/tex]
[tex]\[ 0.866 \neq 4.33 \, (option 2) \][/tex]
[tex]\[ 0.866 \neq 1.414 \, (option 3) \][/tex]
[tex]\[ 0.866 \neq 8.66 \, (option 4) \][/tex]
Since none of the provided options match the calculated height [tex]\(\frac{\sqrt{3}}{2} s\)[/tex], it suggests that there is either an error in the given choices, or the true answer is not provided among the options.
After careful evaluation and comparison, none of the given options accurately represent the height of the triangular base of the pyramid based on the correct geometric formula for an equilateral triangle.
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