Find the length of the height of an equilateral triangle of side 24 cm. `[Take sqrt3 = 1.73]`

Answer

To find the height of an equilateral triangle with a side length of 24 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 height can be found using the Pythagorean theorem.

### Step 2: Draw the triangle and label it
Let’s label the triangle as ABC, where AB = AC = BC = 24 cm. We will drop a perpendicular from point B to the midpoint D of side AC. This creates two right triangles, ABD and CBD.

### Step 3: Determine the lengths of the segments
Since D is the midpoint of AC, we have:
- AD = DC = 24 cm / 2 = 12 cm.

### Step 4: Apply the Pythagorean theorem
In triangle ABD, we can apply the Pythagorean theorem:
\[ AB^2 = AD^2 + BD^2 \]
Where:
- AB = 24 cm (the side of the triangle),
- AD = 12 cm (half the base),
- BD = height (which we need to find).

### Step 5: Substitute the known values into the equation
Substituting the known values into the equation:
\[ 24^2 = 12^2 + BD^2 \]

### Step 6: Calculate the squares
Calculating the squares:
- \( 24^2 = 576 \)
- \( 12^2 = 144 \)

So the equation becomes:
\[ 576 = 144 + BD^2 \]

### Step 7: Rearrange the equation to solve for BD^2
Rearranging gives:
\[ BD^2 = 576 - 144 \]
\[ BD^2 = 432 \]

### Step 8: Take the square root to find BD
Now, take the square root of both sides to find BD:
\[ BD = \sqrt{432} \]

### Step 9: Simplify \(\sqrt{432}\)
To simplify \(\sqrt{432}\):
\[ \sqrt{432} = \sqrt{144 \times 3} = \sqrt{144} \times \sqrt{3} = 12\sqrt{3} \]

### Step 10: Substitute the value of \(\sqrt{3}\)
Now, substituting the value of \(\sqrt{3} = 1.73\):
\[ BD = 12 \times 1.73 = 20.76 \text{ cm} \]

### Final Answer
The height of the equilateral triangle is approximately **20.76 cm**.

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