How to calculate triangle MNP ?

enter image source here
Know S triangle ABC is 24 cm^2
In AB, take M, know AM = MB.
In BC, take N, know BN = 1/3 BC
In the long stretch of AC, take P, know CP = 1/4 AC.

Calculate S triangle MNP.

2 Answers
CW
Mar 31, 2018

area of DeltaMNP=9 " cm"^2

Explanation:

enter image source here
Let |XYZ| denote area of DeltaXYZ
let |MNP|= yellow area, and |NBP|= purple area
|MNP|=|ABP|-|NBP|-|MBN|-|AMP|
given |ABC|=24 " cm"^2,
as CP=1/4AC, => |CBP|=1/4|ABC|=1/4*24=6
as BN=1/3BC, => |NBP|=1/3|CBP|=1/3*6=2
=> |ABP|=|ABC|+|CBP|=24+6=30
as AM=MB, => |AMP|=1/2|ABP|=1/2*30=15
as BM=AM, => |MBC|=1/2|ABC|=1/2*24=12
as BN=1/3BC, => |MBN|=1/3|MBC|=1/3*12=4

Now, |MNP|= |ABP|-|NBP|-|MBN|-|AMP|
=30-2-4-15=9 " cm"^2

CW
Apr 4, 2018

Area of DeltaMNP=9 " cm"^2

Explanation:

Solution 2 :
enter image source here
enter image source here
Let |XYZ| denote area of DeltaXYZ
see Fig 1
given |ABC|=24 " cm"^@, and CP=1/4AC,
=> |CBP|=1/4|ABC|=1/4*24=6 " cm"^2
=> color(red)(|ABP|)=|ABC|+|CBP|=24+6=color(red)(30) " cm"^2
as BN=1/3BC, => color(red)(|NBP|)=1/3|CBP|=1/3*6=color(red)(2) " cm"^2
See Fig 2,
as MB=1/2AB, => |MBC|=1/2|ABC|=1/2*24=12 " cm"^2
as BN=1/3BC, => color(red)(|MBN|)=1/3|MBC|=1/3*12=color(red)(4) " cm"^2
See Fig 3,
as AM=1/2AB, => color(red)(|AMP|)=1/2|ABP|=1/2*30=color(red)(15) " cm"^2

Now, |MNP|=" orange area "=|ABP|-|NBP|-|MBN|-|AMP|
=30-2-4-15=9 " cm"^2

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