Triangle A has sides of lengths #1 3 ,1 4#, and #1 8#. Triangle B is similar to triangle A and has a side of length #4 #. What are the possible lengths of the other two sides of triangle B?

Since the triangles are similar, that means that the side lengths have the same ratio, i.e. we can multiply all of the lengths and get another. For example, an equilateral triangle has side lengths (1, 1, 1) and a similar triangle might have lengths (2, 2, 2) or (78, 78, 78), or something similar. An isosceles triangle may have (3, 3, 2) so a similar may have (6, 6, 4) or (12, 12, 8).

So here we start with (13, 14, 18) and we have three possibilities: (4, ?, ?) , (?, 4, ?), or (?, ?, 4). Therefore, we ask what the ratios are.

The possible lengths of the other two sides of triangle B are 12 and 32.

Since Triangle B is similar to Triangle A, the ratios of corresponding sides of Triangle B to Triangle A will be constant.

Let's denote the lengths of the sides of Triangle B as ( x ) and ( y ), where ( x ) is the side corresponding to the side of length 4 in Triangle A.

We can set up proportions using the ratios of corresponding sides:

[ \frac{x}{1} = \frac{4}{3} ] [ \frac{y}{1} = \frac{4}{8} ]

Solving these proportions gives:

[ x = \frac{4}{3} \times 1 = \frac{4}{3} ] [ y = \frac{4}{8} \times 1 = \frac{1}{2} ]

Thus, the possible lengths of the other two sides of Triangle B are ( \frac{4}{3} ) and ( \frac{1}{2} ).