Question

# The midpoints of the sides of a triangle along with any of the vertices as the fourth point makes a parallelogram of area equal to (a) (b) (c) (d) $\mathrm{ar}\left(∆ABC\right)$

Solution

## (a) $\frac{1}{2}$ ( ar ∆ ABC) Join, FE. ∆ABC has been divided into 4 triangles of equal areas.  So, ​ar(∆AFE) = $\frac{1}{4}$​× ​(ar∆ABC) ∴ ar(∣​∣gm AFDE) = ar (∆AFE) + ar(∆FED)                             =  2 × ​ar(∆AFE) = ​2 × ​$\frac{1}{4}$​ × ​(ar∆ABC) = ​$\frac{1}{2}$(ar∆ABC) Hence, ​ar(∣​∣gm AFDE) = ​​$\frac{1}{2}$(ar∆ABC)MathematicsRS Aggarwal (2017)Standard IX

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