To find the length of the altitude of an equilateral triangle with a side length of 4 centimeters and express it in inches, we can follow these detailed steps:
1. Calculate the altitude in centimeters:
- In an equilateral triangle, the altitude (height) can be calculated using the formula:
\(\[
\text{altitude} = \frac{\text{side length} \times \sqrt{3}}{2}
\]\)
- Given the side length \(\( s = 4 \)\) cm, the altitude (h) will be:
\(\[
h = \frac{4 \times \sqrt{3}}{2} = 2\sqrt{3} \text{ cm}
\]\)
2. Convert the altitude from centimeters to inches:
- We know that \(\( 1 \)\) cm is approximately \(\( 0.393701 \)\) inches.
- The altitude in centimeters is \(\( 2\sqrt{3} \)\) cm. To convert this to inches:
\(\[
\text{altitude in inches} = 2\sqrt{3} \times 0.393701
\]\)
- We can approximate this calculation (without using a calculator):
\(\[
\sqrt{3} \approx 1.732
\]\)
\(\[
2 \sqrt{3} \approx 2 \times 1.732 = 3.464 \text{ cm}
\]\)
\(\[
3.464 \times 0.393701 \approx 1.364 \text{ inches}
\]\)
3. Express altitude in inches as a simplified radical:
- To write the altitude in inches as a simplified radical:
- Initially, we converted \(\( 2\sqrt{3} \)\) cm directly into inches using the conversion factor.
\(\[
2\sqrt{3} \text{ cm} \times \frac{1 \text{ inch}}{2.54 \text{ cm}/\text{inch}} = \frac{2\sqrt{3}}{2.54} \text{ inches}
\]\)
- Simplifying this fraction:
\(\[
\frac{2\sqrt{3}}{2.54} = \frac{2\sqrt{3}}{2 \times 1.27} = \frac{\sqrt{3}}{1.27} \text{ inches}
\]\)
So, the length of the altitude in inches, expressed as a simplified radical, is:
\(\[
\frac{\sqrt{3}}{1.27} \text{ inches}
\]\)
