Question

In \(\Delta ABC\), \(D\) lies on \(AB\), \(\left| {CD} \right| = 5{\rm{ cm}}\), \(\left| {AD} \right| = 4{\rm{ cm}}\), \(\left| {BD} \right| = 4{\rm{ cm}}\), and \({\rm{CD}} \bot {\rm{AB}}\). Where should a point \(P\) be chosen on \(CD\) so that the sum \(\left| {PA} \right| + \left| {PB} \right| + \left| {PC} \right|\) is a minimum? What if \(\left| {CD} \right| = 2{\rm{ cm}}\)?

Short answer

Thus, the minimum value of sum is \(11.93\).

Step-by-step solution

Draw the figure according to the question.

As per the given information, draw a diagram as follow:

\(\begin{aligned}{c}\left| {PA} \right| + \left| {PB} \right| + \left| {PC} \right| = 2\sqrt {{x^2} + 16} + \left( {5 - x} \right)\\Sum = 2\sqrt {{x^2} + 16} + \left( {5 - x} \right)\end{aligned}\)

Consider DC is 5, then \(DP = x\) and \(PC = 5 - x\).

Hypotenuse PB is solved as follows:

\(\begin{aligned}{c}PB = \sqrt {{{\left( {DP} \right)}^2} + {4^2}} \\ = \sqrt {{x^2} + 16} \end{aligned}\)

Similarly, Hypotenuse PA will be same.

Thus, write as follows:

Differentiate the given sum of the sides.

Differentiate the above expression gives:

\(2\frac{1}{{2\sqrt {{x^2} + 16} }}\left( {2x} \right) - 1 = \frac{{2x}}{{\sqrt {{x^2} + 16} }} - 1\)

Put the above expression equal to zero and solve as follows:

\(\begin{aligned}{c}\frac{{2x}}{{\sqrt {{x^2} + 16} }} - 1 = 0\\2x = \sqrt {{x^2} + 16} \\4{x^2} = {x^2} + 16\\3{x^2} = 16\\{x^2} = \frac{{16}}{3}\\x = \pm \frac{4}{{\sqrt 3 }}\end{aligned}\)

Substitute the above value in the PB gives,

\(\begin{aligned}{c}PB = \sqrt {{x^2} + 16} \\ = \sqrt {\frac{{16}}{3} + 16} \\ = \frac{8}{{\sqrt 3 }}\end{aligned}\)

We know that, \(PB = PA = \frac{8}{{\sqrt 3 }}\).

Now, the value PC is calculated as follows:

\(\begin{aligned}{c}PC = 5 - x\\ = 5 - \frac{4}{{\sqrt 3 }}\\ = 2.69\end{aligned}\)

Obtain maximum value of sum of the sides.

Thus, the value of \(\left| {PA} \right| + \left| {PB} \right| + \left| {PC} \right|\) is as follows:

\(\begin{aligned}{c}\left| {PA} \right| + \left| {PB} \right| + \left| {PC} \right| = \frac{8}{{\sqrt 3 }} + \frac{8}{{\sqrt 3 }} + 2.69\\ = \frac{{16}}{{\sqrt 3 }} + 2.69\\ = 9.24 + 2.69\\ = 11.93\end{aligned}\)

Hence, the minimum value of sum is \(11.93\).