The area of quadrilateral BCDE = 20.4 sq. units
Let AD = x and AE = y.
Since ΔABC and ΔAED are similar right angled triangles, we have that
AC/AD = AB/AE
AC = AD + CD
= x + 3.
Also, AB = AE + EB
= y + 4
So, AC/AD = AB/AE
(x + 3)/x = (y + 4)/y
Cross-multiplying, we have
y(x + 3) = x(y + 4)
Expanding the brackets, we have
xy + 3y = xy + 4x
3y = 4x
y = 4x/3
In ΔAED, AD² + AE² = DE².
So, x² + y² = 6²
Substituting y = 4x/3 into the equation, we have
x² + y² = 6²
x² + (4x/3)² = 6²
x² + 16x²/9 = 36
(9x² + 16x²)/9 = 36
25x²/9 = 36
Multiplying both sides by 9/25, we have
x² = 36 × 9/25
Taking square root of both sides, we have
x = √(36 × 9/25)
x = 6 × 3/5
x = 18/5
x = 3.6
Since y = 4x/3,
Substituting x into the equation, we have
y = 4 × 3.6/3
y = 4.8
To find the area of quadrilateral BCDE, we subtract the area of ΔAED from area of ΔABC.
So, area of quadrilateral BCDE = area of ΔABC - area of ΔAED
area of ΔABC = 1/2 AC × AB
= 1/2 (x + 3)(y + 4)
= 1/2(3.6 + 3)(4.8 + 4)
= 1/2 × (6.6)(8.8)
= 1/2 × 58.08
= 29.04 square units
area of ΔAED = 1/2 AD × AE
= 1/2xy
= 1/2 × 3.6 × 4.8
= 1/2 × 17.28
= 8.64 square units
area of quadrilateral BCDE = area of ΔABC - area of ΔAED
area of quadrilateral BCDE = 29.04 sq units - 8.64 sq units
area of quadrilateral BCDE = 20.4 sq. units
So, the area of quadrilateral BCDE = 20.4 sq. units
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