Davida bought a pair of earrings, each in the shape of an isosceles triangle with equal side lengths of 24 mm and a base length of 16 mm. she used one earring to make another pair of earrings, similar in shape, with the base length equal to 11 mm. what is the length of the equal-length sides of this new pair of earrings?

The question has two similar triangles. The triangles are isosceles. 
In order for us to get the length of the sides of the other triangle, we can use ratio and proportion.

Triangle 1
side lengths = 24 mm 
base length = 16 mm

Triangle 2
side lengths = ?
base length = 11 mm

24 : 16 = X : 11
24 / 16 = X / 11
16X = 264
X = 16.5 mm

So the length of the equal-length side of the other triangle.