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Answer:
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.
- ΔABC has a right angle at A
- ΔFDE has a right angle at <FDE
- ΔGEC has a right angle at <GEC
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
