`A_(xy),_(yz),A_(zx)` be the area of projections oif asn area a o the xy,yz and zx and planes resepctively, then `A^2=A^2_(xy)+A^2_(yz)+a^2_(zx)`

Options

A. `A_(xy)^2+A_(yz)^2+A_(zx)^2`
B. `sqrt(A_(xy)^2+A_(yz)^2+A_(zx)^2)`
C. `A_(xy)+A_(yz)+A_(zx)`
D. `sqrt(A_(xy)+A_(yz)+A_(zx))`

Answer

To prove the equation \( A^2 = A_{xy}^2 + A_{yz}^2 + A_{zx}^2 \), we will follow a systematic approach using the concept of direction cosines.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have an area \( A \) and its projections on the three coordinate planes: \( A_{xy} \) (projection on the xy-plane), \( A_{yz} \) (projection on the yz-plane), and \( A_{zx} \) (projection on the zx-plane). We need to prove the relationship between these areas.

**Hint**: Recall that projections of an area onto a plane can be related to the angles between the area and the axes.

2. **Define Direction Cosines**:
Let \( \cos \alpha \), \( \cos \beta \), and \( \cos \gamma \) be the direction cosines of the normal to the plane of area \( A \) with respect to the x, y, and z axes, respectively.

**Hint**: Direction cosines are the cosines of the angles between the normal to the plane and the coordinate axes.

3. **Express Projections**:
The projections of the area \( A \) onto the respective planes can be expressed as:
\[ A_{xy} = A \cos \gamma \]
\[ A_{yz} = A \cos \alpha \]
\[ A_{zx} = A \cos \beta \]

**Hint**: Each projection is the area multiplied by the cosine of the angle between the area and the respective axis.

4. **Square the Projections**:
Now, we will square each of these projections:
\[ A_{xy}^2 = (A \cos \gamma)^2 = A^2 \cos^2 \gamma \]
\[ A_{yz}^2 = (A \cos \alpha)^2 = A^2 \cos^2 \alpha \]
\[ A_{zx}^2 = (A \cos \beta)^2 = A^2 \cos^2 \beta \]

**Hint**: Squaring the projections will help us relate them back to the area \( A \).

5. **Add the Squared Projections**:
Now, we add the squared projections:
\[ A_{xy}^2 + A_{yz}^2 + A_{zx}^2 = A^2 \cos^2 \gamma + A^2 \cos^2 \alpha + A^2 \cos^2 \beta \]
Factor out \( A^2 \):
\[ = A^2 (\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma) \]

**Hint**: Look for a relationship among the cosines that might simplify the expression.

6. **Use the Identity for Direction Cosines**:
From the properties of direction cosines, we know:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]

**Hint**: This identity is crucial; it relates the squares of the direction cosines to unity.

7. **Final Expression**:
Substitute the identity into the equation:
\[ A_{xy}^2 + A_{yz}^2 + A_{zx}^2 = A^2 \cdot 1 = A^2 \]

**Hint**: This shows that the sum of the squares of the projections equals the square of the area.

8. **Conclusion**:
Thus, we have proved that:
\[ A^2 = A_{xy}^2 + A_{yz}^2 + A_{zx}^2 \]

**Hint**: This completes the proof and confirms the relationship between the area and its projections.