Question

n △ABC points M and Q are points on the sides AB and BC respectively, so that AM=2MB and BQ=QC. Find area of quadrilateral MBQP if AQ ∩ CM =P and AABC=30 cm2.

Answer 1

Denote the area of quadilateral MBQP as A, the area of the triangle MBP as
`A_1` and the area of the triangle BPQ as
`A_2.`

Note that:

1.

`A_(\triangle CMB)=(1)/(3)A_(\triangle ABC)=(1)/(3)\cdot 30=10\ cm^2.`

2.

`A_(\triangle CPQ)=A_2.`

3.

`A_(\triangle ABQ)=(1)/(2)A_(\triangle ABC)=(1)/(2)\cdot 30=15\ cm^2.`

4.

`A_(\triangle APM)=2A_1.`

Now

`A+A_(\triangle CPQ)=A_(\triangle CMB)\Rightarrow A+A_2=10\ cm^2.`

`A+A_(\triangle AMP)=A_(\triangle ABQ)\Rightarrow A+2A_1=15\ cm^2.`

You get the system of three equations:

`\left\{\begin{array}{l}A=A_1+A_2\\A+A_2=10\\A+2A_1=15\end{array}\right..`

Substitute the first equation into the last two:

`\left\{\begin{array}{l}A_1+A_2+A_2=10\\A_1+A_2+2A_1=15\end{array}\right.\Rightarrow \left\{\begin{array}{l}A_1+2A_2=10\\A_2+3A_1=15\end{array}\right..`

From the first equation
`A_1=10-2A_2` and then

`A_2+3(10-2A_2)=15,\\ \\A_2+30-6A_2=15,\\ \\-5A_2=-15,\\ \\A_2=3\ cm^2.`

Thus,

`A_1=10-2\cdot 3=10-6=4\ cm^2`

and

`A=4+3=7\ cm^2.`

Answer: 7 sq. cm