Question

There is a deep river in a village. The villagers want to measure its breadth without crossing the river as force of current of river- water is very strong. Ramesh a student of class VII of that village came and said “I will measure the breadth of the river without crossing it.” He came on the bank of the river at a point A and imagined a point B just opposite on the other bank. He moves to C and then D such that C is equidistant from A and D. Then he moves to E such that B, C and E are on the same line and DE is perpendicular to the bank of the river. 1. How can it be possible? Explain. 2. If AC = 7 m and CE = 25 m, find the breadth of the river. 3. Explain why AB is parallel to DE? 4. What portion of the land Ramesh used to measure the breadth of the river? ​

Answer 1

answer

twenty-eight meters

Answer 2

Given: $AC=7 m, CE=25 m$.

To explain: why $AB || DE$ and what portion of the land did Ramesh use to measure the breadth of the river.

Solution:

Draw the required figure.

Find the length of DE.

Know that, in $\[\Delta EDC\]$, Apply Pythagoras theorem,

$\[\begin{array}{l}C{E^2} = D{E^2} + C{D^2}\\ \Rightarrow {25^2} = D{E^2} + {7^2}\\ \Rightarrow D{E^2} = 625 - 49\\ \Rightarrow D{E^2} = 576\\ \Rightarrow D{E^2} = 24\end{array}\]$

Check whether $AB || DE$  or not?

Consider $\[\Delta EDC\,and\,\Delta BAC\]$.

$\[\angle BCA = \angle ECD\]$ [ vertically opposite angle]

$AC=CD=7 m$

$\[\angle BAC = \angle EDC = {90^ \circ }\]$

Therefore, by ASA congruency criteria,

$\[\Delta EDC{\mkern 1mu} \cong \Delta BAC\]$

$\[ \Rightarrow AB = DE\left( {by\,\,CPCT} \right)\]$

$\[ \Rightarrow AB||DE\]$

Understand that, Ramesh used $\[\Delta EDC\]$ to measure the breadth of the river which is 24 m.

Hence, the breadth of the river is 24 m.