Can you me find area of the shaded region?

B

27

Explanation:

Step One

The very first step is to show that triangles ΔABC, ΔFDC and ΔGEC are similar to one another.

1. All three triangles have a right angle in them.

2. All three triangles have a common angle at C

3. All three triangles are similar by AA

Step Two

Find the ratios of the heights to one another.

AC / DC = 3/2

AB / DF = 3/2  The sides of similar triangles are in the same ratio.

Step Three

Find the area of ΔABC

Area ΔABC = 1/2 AB * AC

Area ΔABC = 81

Step Four

Find the Area of ΔDFC

Area of ΔDFC = 1/2 DF * DC

But DF and DC are known in terms of AB and AC

Area of ADC = 1/2 * 2/3 * AB * 2/3 * AC

Area of ADC = 1/2 * 4/9 * AB * AC

However 1/2 AB * AC = 81 so

Area ADC = 4/9 * 81 = 36

That's a very long complex step. Make sure you follow it through.

Step Five

Find the area of ΔGEC

By a similar process

EC = 1/3 AC

EG = 1/3 AB

Area ΔGEC = 1/2 * EC * EG

Area ΔGEC = 1/2 * 1/3 AC * 1/3 AB

Area ΔGEC = 1/2 * 1/9 * AB * AC

But 1/2 AB * AC = 81

Area ΔGEC = 1/9 * 81 = 9

Last Step

Find the area of the shaded area

ΔFDC - ΔGEC = Shaded area

36 -9 = 27 = shaded area