A vertical pole of length 15 m standing on the point A. ABC are points on horizontal ground forming a triangle with `angleA = 90^(@)`.
If AB = 6m` and AC = 8m and a point 'D' (on horizontal plane) is equidistant from A, B, C. Find distance of D from top of the pole

Options

A. `5sqrt(10)m`
B. `10sqrt(5)m`
C. `6sqrt(10)m`
D. None of these

Explanation

To solve the problem, we need to find the distance from point D (which is equidistant from points A, B, and C) to the top of the vertical pole at point A. Let's break down the solution step by step.

### Step 1: Understand the triangle ABC
We have a right triangle ABC with:
- Angle A = 90°
- AB = 6 m
- AC = 8 m

Using the Pythagorean theorem, we can find the length of the hypotenuse BC.

### Step 2: Calculate the length of BC
Using the Pythagorean theorem:
\[ BC = \sqrt{AB^2 + AC^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ m} \]

### Step 3: Find the coordinates of points A, B, and C
Assuming point A is at the origin (0, 0):
- Point B can be at (6, 0) since AB = 6 m along the x-axis.
- Point C can be at (0, 8) since AC = 8 m along the y-axis.

Thus, the coordinates are:
- A(0, 0)
- B(6, 0)
- C(0, 8)

### Step 4: Find the midpoint D of hypotenuse BC
The coordinates of point D, which is the midpoint of BC, can be calculated as:
\[ D\left(\frac{x_B + x_C}{2}, \frac{y_B + y_C}{2}\right) = D\left(\frac{6 + 0}{2}, \frac{0 + 8}{2}\right) = D(3, 4) \]

### Step 5: Determine the height of the pole
The height of the vertical pole at point A is given as 15 m. Therefore, the coordinates of the top of the pole (point P) are:
- P(0, 0, 15)

### Step 6: Calculate the distance PD from point D to the top of the pole P
To find the distance PD, we use the 3D distance formula:
\[ PD = \sqrt{(x_P - x_D)^2 + (y_P - y_D)^2 + (z_P - z_D)^2} \]
Substituting the coordinates:
\[ PD = \sqrt{(0 - 3)^2 + (0 - 4)^2 + (15 - 0)^2} \]
\[ PD = \sqrt{(-3)^2 + (-4)^2 + (15)^2} \]
\[ PD = \sqrt{9 + 16 + 225} = \sqrt{250} \]
\[ PD = \sqrt{25 \times 10} = 5\sqrt{10} \text{ m} \]

### Final Answer
The distance from point D to the top of the pole is \(5\sqrt{10}\) meters.

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