Question

If triangles ABC and DCB are such that $ar\left(△ABC\right)=ar\left(△DCB\right)$ as shown below, then AM= ____.

• A. DN
• B. OD
• C. None of the above
• D. ON

Answer 1

The correct option is A DN

Given that $ar\left(△ABC\right)=ar\left(△DCB\right)$, where the triangles lie on the same base BC.
Area of $△ABC=\frac{1}{2}\times BC\times AM$ and
Area of $△DCB=\frac{1}{2}\times BC\times DN$
But $ar\left(△ABC\right)=ar\left(△DCB\right)$
$\frac{1}{2}\times BC\times AM=\frac{1}{2}\times BC\times DN$
$⟹AM=DN$.
In fact, this is stated as the following theorem:
Triangles with same base and equal areas have equal corresponding altitudes.