Short Questions (Question 10)

Question 10 (4 marks):
Diagram 10 shows the position of three campsites A, B and C at a part of a riverbank drawn on a Cartesian plane, such that A and B lie on the same straight riverbank.

Diagram 10

Sam wants to cross the river from campsite C to the opposite riverbank where the campsites A and B are located.
Find the shortest distance, in m, that he can take to cross the river. Give your answer correct to four decimal places.

Solution:

$\begin{array}{l}\text{Let the shortest distance from campsite }C\\ \text{to opposite riverbank is }CD\text{.}\\ \text{Area of }△ABC\\ =\frac{1}{2}|\begin{array}{l}-2\text{ 7 4 }\\ \text{ 3 }5-\text{3 }\end{array}\begin{array}{l}-2\\ \text{ 3}\end{array}|\\ =\frac{1}{2}|\left(-2\right)\left(5\right)+\left(7\right)\left(-3\right)+\left(4\right)\left(3\right)\\ -\left(3\right)\left(7\right)-\left(5\right)\left(4\right)-\left(-3\right)\left(-2\right)|\\ =\frac{1}{2}|-66|\\ =33{\text{ units}}^2\\ \\ \text{Distance of }AB\\ =\sqrt{{\left(-2-7\right)}^2+{\left(3-5\right)}^2}\\ =\sqrt{85}\text{ m}\\ \\ \text{Area of }△ABC=33\\ \frac{1}{2}\left(AB\right)\left(CD\right)=33\\ \frac{1}{2}\left(\sqrt{85}\right)\left(CD\right)=33\\ CD=\frac{33\left(2\right)}{\sqrt{85}}\\ CD=7.1587\left(4\text{ d}\text{.p}\text{.}\right)\\ \\ \text{Thus, the shortest distance is 7}\text{.1587 m}\text{.}\end{array}$