To solve the problem, let's break it down step by step:

Step 1: Understand the Geometry

Given:

• An isosceles right triangle with the hypotenuse $AB$ of length 10 units.
• A rectangle is inscribed within this triangle.

Step 2: Determine the Coordinates of Points

Since $\triangle ABC$ is isosceles and right-angled, and the hypotenuse $AB$ has a length of 10 units:

• Let's assume that the triangle is positioned with $A$ at the origin $(0,0)$, $B$ at $(10, 0)$, and $C$ at $(0, 10)$.

Step 3: Equation of Line $AB$

To express the y-coordinate of point $P$ in terms of the x-coordinate, we first need the equation of the line $AB$.

Since the hypotenuse $AB$ lies on a straight line, the slope of the line $AB$ can be found using points $A(0,0)$ and $B(10, 0)$: $\text{slope of } AB = \frac{0 - 10}{10 - 0} = -1$

Thus, the equation of line $AB$ is: $y = -x + 10$

Step 4: Express the y-Coordinate of Point $P$ in Terms of $x$

Let $P(x, y)$ be a point on the line $AB$ such that it lies on the hypotenuse. The y-coordinate of $P$ in terms of $x$ can be written directly using the equation of the line: $y = -x + 10$

Thus, the y-coordinate of point $P$ is $y = -x + 10$.

Would you like to dive deeper into this problem or have any other questions? Here are some related questions to explore further:

1. How do you find the dimensions of the inscribed rectangle using the triangle's properties?
2. What would be the area of the rectangle in terms of $x$?
3. Can you express the x-coordinate of the other corner of the rectangle in terms of $y$?
4. What is the maximum area that the inscribed rectangle can achieve?
5. How does the position of point $P$ affect the dimensions of the inscribed rectangle?

Tip: Remember that when dealing with triangles, especially isosceles ones, symmetry can greatly simplify calculations.