Area of Quadrilateral EICD in a Square Using Coordinate Geometry

Given a square ABCD and points E and F as midpoints of sides AD and AB, respectively, we will explore the technique to find the area of quadrilateral EICD where I is the intersection of lines CF and BE. This exercise combines classic geometry with coordinate geometry to determine the desired area.

1. Defining the Square and Points

We start by defining our square and midpoints:

The square ABCD has vertices: A(0, 1), B(1, 1), C(1, 0), D(0, 0) E is the midpoint of AD, thus E(0.5, 0.5) F is the midpoint of AB, thus F(0.5, 1)

2. Finding the Equations of Lines CF and BE

The next step involves determining the equations of lines CF and BE using the points and slopes:

2.1. Line CF

Points: C(1, 0) and F(0.5, 1) Slope of line CF:

slope frac{1 - 0}{0.5 - 1} frac{1}{-0.5} -2

Using point-slope form from point C(1, 0):

y - 0 -2(x - 1) Rightarrow y -2x 2

2.2. Line BE

Points: B(1, 1) and E(0.5, 0.5) Slope of line BE:

slope frac{0.5 - 1}{0.5 - 1} frac{-0.5}{-0.5} 0.5

Using point-slope form from point B(1, 1):

y - 1 0.5(x - 1) Rightarrow y 0.5x 0.5

3. Finding the Intersection Point I of Lines CF and BE

Setting the equations of CF and BE equal to find the intersection point I involves solving the system of linear equations:

-2x 2 0.5x 0.5

Solving for x and then substituting back to find y):

Solve for x: (frac{-2.5x}{-2.5} frac{-1.5}{-2.5} Rightarrow x frac{3}{5}) Substituting x frac{3}{5} into one of the equations to find y:

y -2left(frac{3}{5}right) 2 -frac{6}{5} 2 frac{4}{5} Thus, the coordinates of point I are Ileft(frac{3}{5}, frac{4}{5}right)right

4. Area of Quadrilateral EICD Using the Shoelace Theorem

We now use the vertices of quadrilateral EICD to calculate its area using the Shoelace Formula:

Vertices: E(0, 0.5), Ileft(frac{3}{5}, frac{4}{5}right)right, C(1, 0), D(0, 0) The formula for the area is:

Substituting the coordinates into the formula:

Area frac{1}{2} left[ 0 cdot frac{4}{5} frac{3}{5} cdot 0 1 cdot 0 0 cdot 0.5 - (0.5 cdot frac{3}{5} frac{4}{5} cdot 1 0 cdot 0 0 cdot 0) right] Area frac{1}{2} left[ 0 0 0 0 - left(frac{1.5}{5} frac{4}{5}right) right] Area frac{1}{2} left[ -left(frac{1.5 4}{5}right) right] frac{1}{2} left[ -frac{5.5}{5} right] frac{11}{20}

The area of quadrilateral EICD is boxed{frac{11}{20}}.