Too little info?
Geometry
Level
4
Find the area of rhombus ABCD given that the radii of the circles circumscribed around triangles ABD and ACD are 12.5 cm and 25 cm respectively.
The answer is 400.
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1 solution
ABCD is a rhombus.
Radius of the circumcircle of ABD is 12.5 (Given)
Let the Diagonals of the rhombus be 2a and 2b
Formula for Circumradius of a triangle is (a b c)/4A Where a b and c are the sies of the triangle and A is area of the triangle.
Hence in triangle ABD,
12.5= 2a sqrt(a^2+b^2) sqrt(a^2+b^2) [Half of the diagonals]/4 1/2 a b 2
=(2a a^2+b^2)/4 a*b
=a^2+b^2/2b=12.5cm
Similarly for the second triangle, ACD
= 2b sqrt(a^2+b^2) sqrt(a^2+b^2)/4 a b
=25=a^2+b^2/2a
Therefor 50a=a^2+b^2=25b.
Hence implying, b=2a.
Substituting,
50a= a^2+(2a)^2
=5oa=5a^2
a=10.
Since b=2a
b=20.
Area of the rhombus= 1/2 d1 d2
=1/2 2a 2b
1/2 20 40
=20*20
=400. Ans