Find the length of a altitude in on equilateral triangle of side 'a' cm

Answer

To find the length of the altitude in an equilateral triangle with side length 'a' cm, we can follow these steps:

### Step 1: Understand the properties of the equilateral triangle
An equilateral triangle has all three sides equal, and the altitude (height) drawn from one vertex to the opposite side bisects that side at a right angle.

### Step 2: Draw the triangle and label it
Let’s draw an equilateral triangle ABC where each side is 'a' cm. The altitude AD is drawn from vertex A to the midpoint D of side BC.

### Step 3: Identify the segments
Since D is the midpoint of BC, we can say that:
- BD = DC = a/2 (because D divides BC into two equal halves).

### Step 4: Apply the Pythagorean theorem
In triangle ABD, we have:
- AB = a (hypotenuse),
- AD = h (the altitude we want to find),
- BD = a/2 (one leg of the right triangle).

According to the Pythagorean theorem:
\[ AB^2 = AD^2 + BD^2 \]

### Step 5: Substitute the known values
Substituting the values we have:
\[ a^2 = h^2 + \left(\frac{a}{2}\right)^2 \]

### Step 6: Simplify the equation
Calculating \(\left(\frac{a}{2}\right)^2\):
\[ \left(\frac{a}{2}\right)^2 = \frac{a^2}{4} \]

Now substituting this back into the equation:
\[ a^2 = h^2 + \frac{a^2}{4} \]

### Step 7: Rearrange the equation
To isolate \(h^2\), we can rearrange the equation:
\[ h^2 = a^2 - \frac{a^2}{4} \]

### Step 8: Find a common denominator
Finding a common denominator for the right-hand side:
\[ h^2 = \frac{4a^2}{4} - \frac{a^2}{4} = \frac{3a^2}{4} \]

### Step 9: Take the square root
To find \(h\), we take the square root of both sides:
\[ h = \sqrt{\frac{3a^2}{4}} \]

### Step 10: Simplify the expression
This simplifies to:
\[ h = \frac{\sqrt{3}a}{2} \]

### Conclusion
Thus, the length of the altitude (height) of the equilateral triangle is:
\[ h = \frac{\sqrt{3}}{2} a \text{ cm} \]

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