In a trapezium ABCD , side BC is parallel to AD.Also the lengths of side AB,BC,CD and AD are 8,2,8 and 10 units respectively.Find the radius of the circle that passes through all four points A,B,C and D?

1 Answer
Feb 28, 2018

color (blue)(R=2sqrt7)

Explanation:

The following is a figure for the question. So the whole thing looks like this one below.figurefigure

Lets make some construction
figure with constructionfigure with construction
BE" perpendicular to " AD , CF " perpendicular to " AD
EF=2cm
Rest things are mentioned in figure.
Now applying pythagorean theorem
We get BE=4sqrt3 (if you want me to include that part too, just let me know in comments)

Before proceeding any further i would like to tell that i'll find circumcentre of triangle ABC and because of symmetry that becomes circumcentre of trapezium as well.

Since it is a isoceles trapezium
angle A=180°- angleB
angleA=tan^(-1)(sqrt3)=60°
angleB=120°
Ar triangle ABC=1/2 ×sin120° ×8×2=4sqrt3 cm^2 (there's a lot more easier method also)

Now , here's a relation
"area of "triangle= (a*b*c)/(4R) (where a,b,c are sides of triangle and R is radius of circumcircle)
Using cosine rule AC=2sqrt21

Now again using that relation
4sqrt3=(8*2*2sqrt21)/(4R)
R=2sqrt7 (i tried to show cancelling but preview didn't look any better so no cancelling have been shown).
Hope that helps.