ABCD is a square. Draw an equilateral triangle PBC on side BC considering BC is a base and an equilateral triangle QAC on diagonal AC considering AC is a base. Find the value of
`("Area of " Delta PBC)/("Area of " Delta QAC)`.

Options

A. `(1)/(2)`
B. 1
C. `(1)/(3)`
D. `(1)/(4)`

Answer

To solve the problem, we need to find the areas of triangles PBC and QAC and then calculate the ratio of these areas.

### Step 1: Define the side length of the square
Let the side length of square ABCD be \( A \).

### Step 2: Calculate the length of the diagonal AC
The diagonal \( AC \) of the square can be calculated using the Pythagorean theorem:
\[ AC = \sqrt{A^2 + A^2} = \sqrt{2A^2} = A\sqrt{2} \]

### Step 3: Calculate the area of triangle PBC
Triangle PBC is an equilateral triangle with side length equal to the side of the square, which is \( A \). The area of an equilateral triangle is given by the formula:
\[ \text{Area} = \frac{\sqrt{3}}{4} \times \text{side}^2 \]
Thus, the area of triangle PBC is:
\[ \text{Area of } \Delta PBC = \frac{\sqrt{3}}{4} A^2 \]

### Step 4: Calculate the area of triangle QAC
Triangle QAC is also an equilateral triangle, but its side length is the diagonal \( AC = A\sqrt{2} \). Using the same area formula:
\[ \text{Area of } \Delta QAC = \frac{\sqrt{3}}{4} \times (A\sqrt{2})^2 \]
Calculating this gives:
\[ \text{Area of } \Delta QAC = \frac{\sqrt{3}}{4} \times 2A^2 = \frac{\sqrt{3}}{2} A^2 \]

### Step 5: Calculate the ratio of the areas
Now, we can find the ratio of the areas of triangle PBC to triangle QAC:
\[ \frac{\text{Area of } \Delta PBC}{\text{Area of } \Delta QAC} = \frac{\frac{\sqrt{3}}{4} A^2}{\frac{\sqrt{3}}{2} A^2} \]
Simplifying this expression:
\[ = \frac{\frac{\sqrt{3}}{4}}{\frac{\sqrt{3}}{2}} = \frac{1}{4} \times \frac{2}{1} = \frac{1}{2} \]

### Final Answer
The value of the ratio of the areas is:
\[ \frac{\text{Area of } \Delta PBC}{\text{Area of } \Delta QAC} = \frac{1}{2} \]