Question

if A(-2,1) , B(a,0) , C(4,b ) and ,D (1,2) are the vertices of a || gram ABCD, find the values of a and b . hence find the length of its sides .

Answer 1

Since llgm ABCD and ∆ABC lie on the same base and the same parallels, thus,
​ar(||gmABCD)=2ar(∆ABC)
By using the area formula we have,
​Area of a triangle=½|x1(y​2-y3)+x2(y​3-y1)+x3(y1​-y2)
=½|1(3-2)+2(2+2)+(-3)(-2-3)|
=½|1+8+15|
=½×24=12 square units
Therefore, ar(||gm ABCD)= 2×12 =24 square units


Now,
Using the distance formula we have,
AB=√(x2-x1)2+(y2-y1)2
=√(2-1)2+(3+2)2
=√1+25
=√26 units
We know that,
Area of a parallelogram=height×base
=>ar(llgm ABCD)=height ×AB
=>24=height ×√26
=>height=24/√26
=>height=12√26/13 units. (rationalizing the denominator


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