Question: a)\nx\ny\n12\n15\n20\n\nGraph: A right-triangle-like figure in the center-right; the bottom side is 15 from bottom-left to bottom-right, the right side is vertical and labeled 20, and an interior segment of length 12 goes from the bottom-right vertex to a point on the left slanted side at a right angle, splitting the left slanted side into x (upper part) and y (lower part).
1. **Problem Statement:**
We have a right triangle with legs of lengths $15$ (base) and $20$ (height). Inside the triangle, a segment of length $12$ is drawn from the right angle vertex to the hypotenuse, perpendicular to it, dividing the hypotenuse into two parts labeled $x$ and $y$.
2. **Goal:**
Find the lengths $x$ and $y$ of the two segments into which the hypotenuse is divided.
3. **Step 1: Calculate the hypotenuse length**
Using the Pythagorean theorem:
$$
\text{hypotenuse} = \sqrt{15^2 + 20^2} = \sqrt{225 + 400} = \sqrt{625} = 25
$$
4. **Step 2: Use the property of the altitude to the hypotenuse in a right triangle**
The altitude ($12$) to the hypotenuse divides the hypotenuse into two segments $x$ and $y$ such that:
$$
12^2 = x \times y
$$
which means:
$$
144 = x y
$$
5. **Step 3: Use the relation between the segments and the legs**
Each leg of the triangle is the geometric mean of the hypotenuse segment adjacent to it and the whole hypotenuse:
$$
15^2 = 225 = 25 \times x \implies x = \frac{225}{25} = 9
$$
$$
20^2 = 400 = 25 \times y \implies y = \frac{400}{25} = 16
$$
6. **Step 4: Verify the product**
Check if $x y = 9 \times 16 = 144$ matches $12^2$:
$$
9 \times 16 = 144
$$
This confirms the solution.
**Final answer:**
$$
x = 9, \quad y = 16
$$