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Home » Mathematics » High School Pre-Olympiad (Ages 14+)
 
dissected Triangle, area
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Georg-A.
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#1
dissected Triangle, area
The triangle ABC is dissected into 4 pieces. The area of 3 pieces are known, calculate the area of the fourth piece



PostPosted: Sat Nov 07, 2009 7:21 am  Back to top 
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FantasyLover
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#2
Doesn't belong to Pre-Olympiad.

Solution

Assign mass of 10 to the point F. Since \frac {EF}{FC} = 1, we know that masses of E and C are 5.

Also, since \frac {DF}{FB} = \frac {3}{7}, we know that masses of D and B are 7 and 3, respectively. We have that mass of A is 7 - 5 = 5 - 3 = 2.

From the results above, we have \frac {AE}{EB} = \frac {3}{2}\implies \frac {[AEC]}{[BEC]} = \frac {3}{2}. Since [BEC] = 14, we have [AEC] = 21. However, since [AEFD] = [AEC] - [DFC], the answer is 21 - 3 = 18.

Answer: 18
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Georg-A.
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#3
Thank you, FantasyLover Smile

I wonder if there are other approaches to solve this problem maybe

PostPosted: Sun Nov 08, 2009 4:52 am  Back to top 
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#4
Another approach

Draw the line segment AE. Let us have [ADE] = x. Since \frac {CE}{EF} = 1,we have that [AEF] = 3 + x.

Since \frac {DE}{EB} = \frac {3}{7}, \frac {x}{10 + x} = \frac {3}{7}\implies x = \frac {15}{2}.

[AFED] = 3 + 2x=18.

Answer: 18

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PostPosted: Sun Nov 08, 2009 6:46 am  Back to top 
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