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
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#

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#