We are given a figure with three similar triangles. We are also given a proportion $\frac{c}{a} = \frac{a}{?}$. We need to find which letter from the figure should replace the question mark to make the proportion correct.

1. Problem Description

We are given a figure with three similar triangles. We are also given a proportion $\frac{c}{a} = \frac{a}{?}$. We need to find which letter from the figure should replace the question mark to make the proportion correct.

2. Solution Steps

The three similar triangles are:

1. The large triangle with sides $a$, $b$, and $c+x$.

2. The triangle with sides $a$, $z$, and $y$.

3. The triangle with sides $z$, $b$, and $x$.

From the proportion $\frac{c}{a}$, we observe that $c$ is part of the largest triangle, and $a$ is also part of the largest triangle. Thus, $\frac{c}{a}$ represents the ratio of a side of the largest triangle ($c$ is opposite to the top vertex and $a$ is opposite to the right vertex).

We are given that $\frac{c}{a} = \frac{a}{?}$.

Since $\frac{c}{a}$ is the ratio of sides in the largest triangle, we want $\frac{a}{?}$ to be the ratio of corresponding sides in a similar triangle.

In the smaller triangle with side $z$, we have a ratio $\frac{a}{y}$, so we want to relate side $a$ to another side.

Let's compare the large triangle ($a$, $c+x$) to the smaller triangle ($y$, $a$), where we are given the ratio $\frac{c}{a}$ in the large triangle. The ratio is the side opposite the top angle to the side opposite the right angle. Then the corresponding sides in the other triangle would be $\frac{a}{y}$, since $a$ corresponds to $c$ and $y$ corresponds to $a$.

However, that does not match.

Consider the ratio of the longer leg ($c+x$) to the leg opposite to the bottom angle ($a$). In the largest triangle, this is $\frac{c+x}{a}$. In the similar triangle ($a$, $y$), this corresponds to $\frac{a}{z}$. Since these sides of the two triangles correspond, we must have

$\frac{c+x}{a}=\frac{a}{z}$

Since $a$ is to the right of $y$, it means that the missing part is $y$.

Looking at the middle-sized triangle, we have sides $a$, $y$ and $z$.

$\frac{c}{a}$ corresponds to the ratio of hypotenuse to the side $a$. The hypotenuse is formed by $x+c$.

Consider the similarity of the big triangle and the triangle with sides $a,y,z$.

If we consider that $c$ is the hypotenuse and $a$ is one of the legs, then,

$\frac{c}{a} = \frac{a}{y}$

$\frac{\text{hypotenuse}}{\text{leg}} = \frac{\text{leg}}{\text{corresponding leg}}$

Therefore, the answer is $y$.

3. Final Answer

y