Let WXYZ be a square. Let P, Q, R, be the midpoints of WX, XY and ZW respectively and K, L be the midpoints of PQ and PR respectively. What is the value of
`"Area of triangle PKL"/"Area of square WXYZ"=?`

Options

A. `1/32`
B. `1/16`
C. `1/8`
D. `1/64`

Answer

To find the ratio of the area of triangle PKL to the area of square WXYZ, we can follow these steps:

### Step 1: Define the Square and Midpoints
Let the square WXYZ have a side length of \( s \). The coordinates of the vertices can be defined as:
- \( W(0, 0) \)
- \( X(s, 0) \)
- \( Y(s, s) \)
- \( Z(0, s) \)

Next, we find the midpoints:
- \( P \) is the midpoint of \( WX \): \( P\left(\frac{s}{2}, 0\right) \)
- \( Q \) is the midpoint of \( XY \): \( Q\left(s, \frac{s}{2}\right) \)
- \( R \) is the midpoint of \( ZW \): \( R\left(0, \frac{s}{2}\right) \)

### Step 2: Find Midpoints K and L
Now we find the midpoints \( K \) and \( L \):
- \( K \) is the midpoint of \( PQ \):
\[ K\left(\frac{\frac{s}{2} + s}{2}, \frac{0 + \frac{s}{2}}{2}\right) = K\left(\frac{3s}{4}, \frac{s}{4}\right) \]
- \( L \) is the midpoint of \( PR \):
\[ L\left(\frac{\frac{s}{2} + 0}{2}, \frac{0 + \frac{s}{2}}{2}\right) = L\left(\frac{s}{4}, \frac{s}{4}\right) \]

### Step 3: Calculate the Area of Triangle PKL
The area of triangle PKL can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Substituting the coordinates:
- \( P\left(\frac{s}{2}, 0\right) \)
- \( K\left(\frac{3s}{4}, \frac{s}{4}\right) \)
- \( L\left(\frac{s}{4}, \frac{s}{4}\right) \)

The area becomes:
\[ \text{Area} = \frac{1}{2} \left| \frac{s}{2}\left(\frac{s}{4} - \frac{s}{4}\right) + \frac{3s}{4}\left(\frac{s}{4} - 0\right) + \frac{s}{4}\left(0 - \frac{s}{4}\right) \right| \]
\[ = \frac{1}{2} \left| 0 + \frac{3s^2}{16} - \frac{s^2}{16} \right| = \frac{1}{2} \left| \frac{2s^2}{16} \right| = \frac{s^2}{16} \]

### Step 4: Calculate the Area of Square WXYZ
The area of square WXYZ is:
\[ \text{Area}_{WXYZ} = s^2 \]

### Step 5: Find the Ratio of Areas
Now, we can find the ratio of the area of triangle PKL to the area of square WXYZ:
\[ \text{Ratio} = \frac{\text{Area}_{PKL}}{\text{Area}_{WXYZ}} = \frac{\frac{s^2}{16}}{s^2} = \frac{1}{16} \]

### Final Answer
Thus, the value of \( \frac{\text{Area of triangle PKL}}{\text{Area of square WXYZ}} \) is \( \frac{1}{16} \).

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