In a square PQRS, X and Y are mid points of sides PS and QR respectively. XY and QS intersect a t O. Find the area of `Delta`XOS, if PQ = 8 cm.

Options

A. 6 `cm^(2)`
B. 12 `cm^(2)`
C. 4 `cm^(2)`
D. 8 `cm^(2)`

Answer

To find the area of triangle XOS in square PQRS where PQ = 8 cm, we can follow these steps:

### Step 1: Draw the Square
Draw square PQRS with vertices P, Q, R, and S. Since PQ = 8 cm, each side of the square is also 8 cm.

### Step 2: Identify Midpoints
Identify the midpoints X and Y of sides PS and QR respectively. Since PS and QR are both 8 cm, the coordinates of the midpoints will be:
- X (midpoint of PS) = (0, 4)
- Y (midpoint of QR) = (8, 4)

### Step 3: Determine Coordinates of Vertices
Assign coordinates to the vertices of the square:
- P = (0, 0)
- Q = (8, 0)
- R = (8, 8)
- S = (0, 8)

### Step 4: Find the Equation of Line XY
The line segment XY is horizontal at y = 4, as both X and Y have the same y-coordinate. Therefore, the equation of line XY is:
\[ y = 4 \]

### Step 5: Find the Equation of Line QS
The line segment QS connects points Q and S. The slope of line QS can be calculated as:
\[ \text{slope} = \frac{8 - 0}{0 - 8} = -1 \]
Using point-slope form, the equation of line QS is:
\[ y - 0 = -1(x - 8) \]
Simplifying this gives:
\[ y = -x + 8 \]

### Step 6: Find the Intersection Point O
To find the intersection point O of lines XY and QS, set their equations equal:
\[ 4 = -x + 8 \]
Solving for x gives:
\[ x = 4 \]
Thus, the coordinates of point O are:
\[ O = (4, 4) \]

### Step 7: Determine the Area of Triangle XOS
Triangle XOS has vertices at X(0, 4), O(4, 4), and S(0, 8). The base of triangle XOS can be taken as segment XO, which is 4 cm (from (0, 4) to (4, 4)), and the height from point S to line XO, which is 4 cm (from y = 4 to y = 8).

### Step 8: Calculate the Area
The area of triangle XOS can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Substituting the values:
\[ \text{Area} = \frac{1}{2} \times 4 \times 4 = \frac{16}{2} = 8 \, \text{cm}^2 \]

Thus, the area of triangle XOS is **8 cm²**.

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