# Find b, c and d so that the quadrilateral is a parallelogram with area equal to 80 square units?

Nov 7, 2016

$b = 14 , c = 8 , d = 18$

#### Explanation:

To achieve parallelism

${y}_{C} - {y}_{B} = {y}_{D} - {y}_{A}$

$d - b = 2 - \left(- 2\right)$

$d = b + 4$

${x}_{C} - {x}_{B} = {x}_{D} - {x}_{A}$

$c - 2 = 4 - \left(- 2\right)$

$c = 2 + 6 = 8$

so

C=(8;b+4)

Now let ${B}^{'} , {D}^{'} , {C}^{'}$ be the projection of $B , D , C$ on the horizontal line passing per A then

${A}_{A B C D} = {A}_{A B {B}^{'}} + {A}_{{B}^{'} B C {C}^{'}} - {A}_{A D {D}^{'}} - {A}_{{D}^{'} D C {C}^{'}}$

so

$80 = \frac{4 \cdot \left(b + 2\right)}{2} + \frac{\left(b + 2 + b + 6\right) \cdot 6}{2} - \frac{6 \cdot 4}{2} - \frac{\left(4 + b + 6\right) \cdot 4}{2}$

$80 = 2 b + 4 + 6 b + 24 - 12 - 2 b - 20$

$80 = 6 b - 4$

$6 b = 84$

$b = 14$

$d = 18$