Question

If the areas of two triangles are in the ratio of $m:n$ and the ratio of their bases is $p:q,$ then the ratio of their altitudes is

• A.
  $mp:nq$
• B.
  $mq:np$
• C.
  $m+n:p+q$
• D.
  $m+p:n+q$

Answer 1

The correct option is B

$mq:np$
Let areas of the two triangles be $A_{1}and{A}_2,$ bases be $b_{1}and{b}_2$ and altitudes be $h_{1}and{h}_2$ respectively.

$\therefore A_1=\frac{1}{2}{b}_1{h}_1......\left(i\right)and,A_2=\frac{1}{2}{b}_2{h}_2....\left(ii\right)\text{Dividing (i) and (ii), we get}\frac{A_1}{A_2}=\frac{\frac{1}{2}{b}_1{h}_1}{\frac{1}{2}{b}_2{h}_2}\Rightarrow \frac{m}{n}=\frac{p}{q}\times \frac{h_1}{h_2}\Rightarrow \frac{h_1}{h_2}=\frac{mq}{np}\Rightarrow h_1:h_2=mq:np(\because \frac{A_1}{A_2}=\frac{m}{n}and\frac{b_1}{b_2}=\frac{p}{q})$

Hence, the correct answer is option (b).