In the figure let $B(20, 0)$ and $C(0, 30)$ lie in x and y axis respectivelly. The angle, $/_ACB= 90°$ . A rectangle DEFG is inscribed in triangle ABC. Given that the area of triangle CGF is 351, calculate the area of the rectangle DEFG?

Question

1744 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-10-08T12:02:00

$468$

$hat(ACB) = pi/2$ so

$bar(AO)cdot bar(OB)=bar(OC)^2$ so

$abs(Delta(ACB)) = 1/2(bar(AO)+bar(OB))bar(OC)$

We know that $abs(Delta(GCF)) = 351$ and

$abs(bar(OB))=20$ and $bar(OC)=30$ then

$abs(bar(AO))=30^2/20=45$ so

$abs(Delta(ACB)) =1/2(45+20)30=975$

We have also

$abs(Delta(ACB))/bar(AB)^2=abs(Delta(GCF))/bar(GF)^2$ then

$bar(GF) = bar(AB)sqrt(abs(Delta(GCF))/abs(Delta(ACB)))=65sqrt(351/975)=65(3/5)$

Calling now $h_1$ the height of $Delta GCF$ we have

$h_1/bar(GF)=bar(OC)/bar(AB)$ and also

$h_2 = bar(OC)-h_1$ so the sought area is

$square = bar(GF) cdot h_2 = bar(GF) cdot bar(OC)(1-bar(GF)/bar(AB)) = 468$

In the figure let B(20, 0)B(20, 0) and C(0, 30) C(0, 30) lie in x and y axis respectivelly. The angle, /_ACB= 90°/_ACB= 90°. A rectangle DEFG is inscribed in triangle ABC. Given that the area of triangle CGF is 351, calculate the area of the rectangle DEFG?