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In Fig. 9.30, D and E are two points o...

In Fig. 9.30, D and E are two points on BC such that `B D=D E=E C . Show that `a r(A B D)=a r(A D E)=a r(A E C)` . Can you now answer the question that you have left in the Introduction of this chapter, whether the field of Budha has been actually divided into three parts of equal area ?

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Let's draw an altitude that will help us to find areas of all three triangles. Let us draw `AL bot BC`.
We know that,`Area of a triangle` = `1/2 xx` `Base` `xx` Altitude
`Area (triangleADE) = 1/2 xx DE xx AL`
`Area (triangleABD) = 1/2 xx BD xx AL`
`Area (triangleAEC) = 1/2 xxEC xx AL`
`DE = BD = EC` (Given)
Thus, `1/2 xx DE xx AL = 1/2 xx BD xx AL = 1/2 xx EC xx AL`
Therefore, `ar (triangleADE) = ar (triangleABD) = ar (triangleAEC)`
Yes, we can now say that the field of Budhia has been actually divided into three parts of equal area.
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