Question

How to calculate triangle MNP ?

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.

Answer 1

area of `DeltaMNP=9 " cm"^2`

Explanation:

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`

Answer 2

Area of `DeltaMNP=9 " cm"^2`

Explanation:

Solution 2 :


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`