Here the two sides AB and AC are equal, so we can conclude that

the${\triangle} ABC$ is isosceles.

Now the minimum length from A to BC will be the perpendicular line.

Since the ${\triangle} ABC$ is isosceles and perpendicular line from A to BC will meet at the midpoint D (as shown in the figure)

Now consider the ${\triangle} ABD$

AB = 100 , BD =80 and we need to find AD

Using Pythagoras theorem

AD = $\sqrt{(10000 - 6400)}$ = 60

or we can see that the sides of ${\triangle} ABD$ are in the ratio 3:4:5

= 60:80:100

Ans = 60