In an equilateral triangle ABC of side 10cm, the side BC is trisected at D. Then the length (in cm) of AD is

Options

A. `3 sqrt(7)`
B. `7 sqrt(3)`
C. `(10 sqrt(7))/(3)`
D. `(7 sqrt(10))/(3)`

Answer

To find the length of AD in the equilateral triangle ABC with side length 10 cm, where side BC is trisected at point D, we can follow these steps:

### Step 1: Understand the triangle and its properties
An equilateral triangle has all sides equal and all angles equal to 60 degrees. In triangle ABC, each side is 10 cm.

### Step 2: Trisect side BC
Since BC is trisected at point D, we can divide BC into three equal parts. Therefore, each segment (BD and DC) will be:
\[ BD = DC = \frac{10 \text{ cm}}{3} \approx 3.33 \text{ cm} \]

### Step 3: Determine the height of the triangle
The height (h) of an equilateral triangle can be calculated using the formula:
\[ h = \frac{\sqrt{3}}{2} \times \text{side} \]
For triangle ABC:
\[ h = \frac{\sqrt{3}}{2} \times 10 = 5\sqrt{3} \text{ cm} \]

### Step 4: Set coordinates for points A, B, and C
We can place the triangle in a coordinate system for easier calculations:
- Let point B be at (0, 0)
- Point C will be at (10, 0)
- Point A will be at \((5, 5\sqrt{3})\)

### Step 5: Determine the coordinates of point D
Since D trisects BC, its coordinates will be:
\[ D = \left(\frac{0 + 10}{3}, 0\right) = \left(\frac{10}{3}, 0\right) \]

### Step 6: Use the distance formula to find AD
The distance AD can be calculated using the distance formula:
\[ AD = \sqrt{(x_A - x_D)^2 + (y_A - y_D)^2} \]
Substituting the coordinates:
- \(A = (5, 5\sqrt{3})\)
- \(D = \left(\frac{10}{3}, 0\right)\)

So,
\[ AD = \sqrt{\left(5 - \frac{10}{3}\right)^2 + \left(5\sqrt{3} - 0\right)^2} \]

### Step 7: Simplify the expression
Calculating \(5 - \frac{10}{3}\):
\[ 5 = \frac{15}{3} \implies 5 - \frac{10}{3} = \frac{15}{3} - \frac{10}{3} = \frac{5}{3} \]
Now substitute back into the distance formula:
\[ AD = \sqrt{\left(\frac{5}{3}\right)^2 + (5\sqrt{3})^2} \]
Calculating each part:
\[ \left(\frac{5}{3}\right)^2 = \frac{25}{9} \]
\[ (5\sqrt{3})^2 = 25 \times 3 = 75 \]
Now combine:
\[ AD = \sqrt{\frac{25}{9} + 75} = \sqrt{\frac{25}{9} + \frac{675}{9}} = \sqrt{\frac{700}{9}} = \frac{\sqrt{700}}{3} \]
Since \(\sqrt{700} = 10\sqrt{7}\):
\[ AD = \frac{10\sqrt{7}}{3} \text{ cm} \]

### Final Answer
The length of AD is:
\[ \frac{10\sqrt{7}}{3} \text{ cm} \]