ABCD is a tetrahedron such that each of the `triangleABC`, `triangleABD` and `triangleACD` has a right angle at A. If `ar(triangleABC) = k_(1). ar(triangleABD)= k_(2), ar(triangleBCD)=k_(3)`, then `ar(triangleACD)` is

Options

A. `sqrt(k_(1)^(2)+k_(2)^(2)+k_(3)^(2))`
B. `sqrt((k_(1)k_(2)k_(3))/(k_(1)+k_(2)+k_(3))`
C. `sqrt(|k_(1)^(2)+k_(2)^(2)-k_(3)^(2)|)`
D. `sqrt(|k_(2)^(2)-k_(1)^(2)-k_(3)^(2)|)`

Explanation

To solve the problem, we need to find the area of triangle ACD given the areas of triangles ABC, ABD, and BCD. Let's denote the areas as follows:

- \( \text{ar}(triangle ABC) = k_1 \)
- \( \text{ar}(triangle ABD) = k_2 \)
- \( \text{ar}(triangle BCD) = k_3 \)
- \( \text{ar}(triangle ACD) = x \) (this is what we need to find)

Since each triangle has a right angle at A, we can use the relationship derived from the Pythagorean theorem in three dimensions.

### Step-by-Step Solution:

1. **Understanding the Right Angles**:
Since triangles ABC, ABD, and ACD have right angles at A, we can apply the Pythagorean theorem in the context of areas. The areas of the triangles can be related through the equation:
\[ \text{ar}(triangle ABC)^2 + \text{ar}(triangle ABD)^2 + \text{ar}(triangle ACD)^2 = \text{ar}(triangle BCD)^2 \]

2. **Substituting the Areas**:
Substitute the known areas into the equation:
\[ k_1^2 + k_2^2 + x^2 = k_3^2 \]

3. **Rearranging the Equation**:
To isolate \( x^2 \), rearrange the equation:
\[ x^2 = k_3^2 - k_1^2 - k_2^2 \]

4. **Taking the Square Root**:
Now, take the square root of both sides to find \( x \):
\[ x = \sqrt{k_3^2 - k_1^2 - k_2^2} \]

5. **Final Result**:
Thus, the area of triangle ACD is given by:
\[ \text{ar}(triangle ACD) = \sqrt{k_3^2 - k_1^2 - k_2^2} \]

### Summary:
The area of triangle ACD can be computed using the areas of the other triangles as follows:
\[ \text{ar}(triangle ACD) = \sqrt{k_3^2 - k_1^2 - k_2^2} \]