BC = √(x² + 2²)
AB = √((12 - x)² + 3²)

The whole idea is to minimise AB + BC. This occurs when the 3 points are collinear and ABC is a straight line.

This means that AB + BC = AC = √(x² + 2²) + √((12 - x)² + 3²)

From the triangle inequality we know that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.

As long as ABC is not a straight line, AB + BC will always be longer than AC.


Now,

From the rectangle, we can look at △ ADC
and use Pythagoras' Theorem.

AD² + DC² = AC²

5² + 12² = AC²

AC² = 169

AC = √169 = 13

So the minimum value is 13.

When AC = 13,

√(x² + 2²) + √((12 - x)² + 3²) = 13

This is actually quite difficult to solve.

But what we realise is that :

The right angled triangles with hypotenuses √(x² + 2²) and √((12 - x)² + 3²) , are actually similar to △ ADC.

(It is easy to prove using AA similarity)


The triangle containing √(x² + 2²) has sides x, 2 and √(x² + 2²)

Since this triangle and△ADC are similar, The ratio of their corresponding sides are equal.


x / DC = 2 / AD

x/12 = 2/5

x = 12 × 2/5

x = 24/5

x = 4.8

thank you