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October 1st, 2011, 09:17 PM   #1
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General Area and Perimeter Calculation

Let ABC be an acute-angled triangle in which D, E, F are points on BC, CA, AB respectively, such that AD is perpendicular to BC, AE=EC and CF bisects angle C. Suppose CF meets AD and DE in M and N respectively. If FM=2, MN=1, NC=3, find the perimeter and area of triangle.

I am having some difficulties in solving this problem. Can you guys help me???
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October 2nd, 2011, 02:38 AM   #2
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Re: General Area and Perimeter CAlculation

Pls help me on this problem people..
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October 3rd, 2011, 07:58 AM   #3
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Re: General Area and Perimeter Calculation

I've got many things here but i cannot reach on area part...:O
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December 24th, 2017, 03:41 AM   #4
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Let ABC be a acute angled triangle in which D,E,F are points on BC,CA,AB respectively

Given FN = NC = 3. In triangle ACF , CE/AE = CN /NF = 1.
So, by midpoint theorem, EN || AF. => DE || AB. and EN = AF/2
In triangle ABC, CE = EA and DE || AB.
So, by intercept theorem BD= DC. => ABC is an isosceles triangle with AB = AC.

Consider triangles DMN and AMF. They are similar to each other by AA similarity criterion.
So, DN/AF = MN/MF = ½. Therefore DN = EN = AF/2.

In triangle CED, by angle bisector theorem CE/CD = EN/DN. But EN = DN.
Therefore CE = CD => AC= BC. So, ABC is an equilateral triangle with altitude CF = 6.

So, perimeter = 3 * AB = 3 * 2* CF/ sqrt(3). = 6* 6/ sqrt (3) = 12* sqrt(3)= 20.76
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December 24th, 2017, 05:44 AM   #5
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Nice solve OMC.
Would have been easier if problem was stated:
Find perimeter of equilateral triangle with height = 6

By the way, 12*sqrt(3) = ~20.78
Since Xmas tomorrow, you're forgiven

NOTE: problem posted over 6 years ago...
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