Sometimes it's best to start by breaking a problem down into pieces and then seeing how the pieces are related. To do that here, we're going to need to introduce some variables. The segment with length $48$ splits the right triangle into two smaller right triangles. Two of their side lengths add up to $100;$ let's call them $a$ and $b.$

That still doesn't tell us very much about how these two triangles are related. We looked at some of their sides; let's now look at their angles. Every triangle's measures add up to $180^\circ.$ Since each of these triangles has a $90^\circ$ angle, its two acute angles' measures add up to $90^\circ.$

But then the upper two acute angles of the two triangles form a right angle, so their measures also add up to $90^\circ.$ What does this mean?

The two triangles have congruent angles, so they are similar. This happens every time you split a right triangle with an altitude through its right angle.

We already knew that two of the triangles' side lengths add up to $100.$ Now that we know the triangles' side lengths are proportional, we have another way of relating them:

$\frac{a}{48} = \frac{48}{b} \implies ab = 48^2.$

We need two numbers with a product of $48^2$ and a sum of $100.$ Let's think about prime factors to help to guess and check our way to a solution. (This will only help if the triangle's sides are integers, which we don't know for sure at this point, but we can hope…)

Since $48 = 2^4 \times 3,$ we know that $48^2 = 2^8 \times 3^2.$ If $a$ and $b$ are integers, they must be made of these prime factors. For example, $a = 2^3 \times 3 = 24$ and $b = 2^5 \times 3 = 96$ have a product of $2^8 \times 3^2.$ Their sum, though, is $24 + 96 = 120,$ which is too large.

The combination of factors that does work is $a = 2^6 = 64$ and $b = 2^2 \times 3^2 = 36,$ since $36 + 64 = 100.$

Finally, we can use the Pythagorean theorem to get the missing side lengths:

$\begin{aligned} 64^2 + 48^2 & = 6400 \\ \sqrt{6400} & = 80 \\\\ 36^2 + 48^2 & = 3600 \\ \sqrt{3600} & = 60. \end{aligned}$

Can you take this a step further to solve the challenge below?