Let ABC be a acute angled triangle in which D,E,F

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

Jatheen · Accepted Answer

Given FN = NC = 3. In triangle ACF , CE/AE = CN /NF = 1.So, by midpoint theorem, EN || AF. => DE || AB. and EN = AF/2In 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

10 grade maths> Let ABC be a acute angled triangle in whi...

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

yubraj panigrahi , 7 Years ago

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10 grade maths> Let ABC be a acute angled triangle in whi...

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

yubraj panigrahi , 7 Years ago

Jatheen

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