Question

If the ratio of altitudes of two triangles are m and n and having equal bases, then the ratio of their areas is equal to

• A.
  $m^2:n^2$
• B.
  $m:n$
• C.
  $\frac{1}{m^2}:\frac{1}{n^2}$
  
• D.
  $2m^2:3n^2$

Answer 1

The correct option is B

$m:n$
Let the two triangles be $\Delta ABCand\Delta PQR$ with their altitudes be AD and PS respectively.


$Ar\left(\Delta ABC\right)=\frac{1}{2}\times BC\times ADAr\left(\Delta PQR\right)=\frac{1}{2}\times QR\times PS\therefore \frac{Ar\left(\Delta ABC\right)}{Ar\left(\Delta PQR\right)}=\frac{\frac{1}{2}\times BC\times AD}{\frac{1}{2}\times QR\times PS}=\frac{BC}{QR}\times \frac{AD}{PS}=\frac{AD}{PS}(\because BC=QR)=\frac{m}{n}(\because \frac{AD}{DS}=\frac{m}{n})$

Hence, the correct answer is option (b).