Question

In the adjoining figure, explain how one can find the breadth of the river without crossing it.

Answer 1

Let AB be the breadth of the river.
M is any point situated on the bank of the river.
Let O be the mid point of BM.
​Moving along perpendicular to point such that A,O and N are in a straight line.
Then MN is the required breadth of the river.

$$\begin{gathered} \mathrm{In}△\mathrm{OBA}\mathrm{and}△\mathrm{OMN},\mathrm{we}\mathrm{have}: \\ \mathrm{OB}=\mathrm{OM}(\mathrm{O}\mathrm{is}\mathrm{midpoint}) \\ \angle \mathrm{OBA}=\angle \mathrm{OMN}(\mathrm{Each}90°) \\ \angle \mathrm{AOB}=\angle \mathrm{NOM}(\mathrm{Vertically}\mathrm{opposite}\mathrm{angle}) \\ \therefore △\mathrm{OBA}\cong △\mathrm{OMN}(\mathrm{ASA}\mathrm{criterion}) \end{gathered}$$

Thus, MN = AB (CPCT)
If MN is known, one can measure the width of the river without actually crossing it.