Let's break down the problem and solve each part step by step.

Problem Recap:

You have a rectangle inscribed in an isosceles right triangle with a hypotenuse of 2626 units. The triangle is positioned with its base along the x-axis, and its hypotenuse lies along the line passing through points $A(-13, 0)$ and $B(13, 0)$.

Part 1: Express the y-coordinate of $P(x, y)$ in terms of $x$.

1. Find the Equation of Line AB:

   • The line $AB$ is the hypotenuse of the isosceles right triangle.
   • The points $A(-13, 0)$ and $B(13, 0)$ determine the equation of the line.
   • The slope of $AB$ is: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2626 - 0}{13 - (-13)} = \frac{2626}{26} = 101 \text{ units}$
   • Since point $B(13, 0)$ lies on the positive y-axis, the y-intercept is 0. The equation of the line AB in slope-intercept form is: $y = 101x$

2. Express the y-coordinate of P(x, y) in Terms of $x$:

   • The y-coordinate of $P(x, y)$ is determined by the line equation $y = 101x$.
   • Thus, the y-coordinate of $P$ is: $y_P = 101x$

Part 2: Express the Area of the Rectangle in Terms of $x$.

1. Dimensions of the Rectangle:

   • The rectangle's height is $y$, which we've determined to be $101x$.
   • The rectangle's base extends from $-x$ to $x$, so the base length is $2x$.

2. Area of the Rectangle:

   • The area $A$ of the rectangle is the product of its base and height. $A = \text{base} \times \text{height} = 2x \times 101x = 202x^2$

Final Answers:

• Part 1 (y-coordinate of $P$): $y_P = 101x$

• Part 2 (Area of the Rectangle): $A = 202x^2$

Would you like further details or have any additional questions?

Here are 5 related questions:

1. What is the general equation for a line given two points?
2. How do the properties of an isosceles triangle help in determining the coordinates of points?
3. What other geometric shapes can be inscribed in a right triangle, and how would their areas be calculated?
4. Can the method used here be generalized for any right triangle with different lengths?
5. How does changing the slope of the line $AB$ affect the area of the rectangle?

Tip: Remember to always check for symmetry and special properties in geometric problems, as these can often simplify your calculations significantly.