To find the area of the green region, we first need to determine the coordinates of point $Q$ and use it to find the area of triangle $DPQ$.

Step 1: Coordinates of Points

Given the square $ABCD$ with points:

• $A(0, 1)$
• $B(1, 1)$
• $C(1, 0)$
• $D(0, 0)$
• $P(0, 0.6)$

We need to find the coordinates of point $Q$, where $Q$ lies on $CD$ and $PQ$ is perpendicular to $BP$.

Step 2: Equation of Line $BP$

The slope of $BP$ can be calculated since $B(1, 1)$ and $P(0, 0.6)$ are given: $\text{slope of } BP = \frac{1 - 0.6}{1 - 0} = \frac{0.4}{1} = 0.4$ Thus, the equation of line $BP$ is: $y - 0.6 = 0.4(x - 0)$ $y = 0.4x + 0.6$

Step 3: Slope of Perpendicular Line $PQ$

The slope of $PQ$ must be the negative reciprocal of $0.4$, which is $-\frac{1}{0.4} = -2.5$.

Step 4: Equation of Line $PQ$

The equation of line $PQ$, passing through $P(0, 0.6)$ with slope $-2.5$, is: $y - 0.6 = -2.5(x - 0)$ $y = -2.5x + 0.6$

Step 5: Intersection of Line $PQ$ with Line $CD$

The line $CD$ is horizontal at $y = 0$ (since $C(1, 0)$ and $D(0, 0)$). Substitute $y = 0$ in the equation of $PQ$: $0 = -2.5x + 0.6$ $2.5x = 0.6$ $x = \frac{0.6}{2.5} = 0.24$ Thus, the coordinates of $Q$ are $(0.24, 0)$.

Step 6: Area of Triangle $DPQ$

The area of triangle $DPQ$ can be calculated using the formula: $\text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height}$ Here, the base $DQ$ is the distance between $D(0, 0)$ and $Q(0.24, 0)$, which is $0.24$. The height is the distance from $P(0, 0.6)$ to line $CD$, which is $0.6$. Therefore: $\text{Area} = \frac{1}{2} \times 0.24 \times 0.6 = \frac{1}{2} \times 0.144 = 0.072$

Thus, the area of the green region is $\boxed{0.072}$ square units.

Would you like more details on any part of the solution or have any other questions?

Here are some related problems:

1. How would you find the area if $P$ were at a different position on $AD$?
2. What is the area of the triangle if $PQ$ were parallel to $AB$ instead of perpendicular to $BP$?
3. How does the area of the green region change if the square is scaled up by a factor of 2?
4. What would be the coordinates of $Q$ if $P$