The given figure shows a rectangle inscribed in an isosceles right triangle whose hypotenuse is10 units long.

Express the​ y-coordinate of P in terms of x.​ (Hint: Write an equation for the line​ AB.)
​(b) Express the area of the rectangle in terms of x.
​(c) What is the largest area the rectangle can​ have, and what are its​ dimensions?

To answer these questions, let's analyze the given figure step by step.

Step 1: Understanding the figure
The figure shows an isosceles right triangle ABC where AB is the hypotenuse. Inside this triangle, there is a rectangle PQRS.

Step 2: Find the equation of the line AB
To find the equation of the line AB, we need to determine its slope and y-intercept. Since AB is a hypotenuse of an isosceles right triangle, its slope is -1 (perpendicular to sides of the triangle). The midpoint of the hypotenuse AB is also the midpoint of the rectangle PQRS.

Step 3: Find the midpoint of AB
To find the midpoint, calculate the average of corresponding x-coordinates and y-coordinates of points A and B.

Step 4: Express the y-coordinate of P in terms of x
The y-coordinate of the midpoint will be the y-coordinate of point P in terms of x since P is the midpoint of side QS of the rectangle. Therefore, write an equation using the midpoint formula and solve for y.

Step 5: Express the area of the rectangle in terms of x
The area of a rectangle is given by the formula A = length * width. Let the length of the rectangle be x and the width be the distance between the y-coordinate of point P and point B. Write an expression for the width and multiply it by x to obtain the area of the rectangle in terms of x.

Step 6: Determine the largest area the rectangle can have and its dimensions.
To find the largest area the rectangle can have, differentiate the area expression obtained in step 5 with respect to x and equate it to zero. Solve for x to find the critical points. The dimensions of the rectangle can then be determined by substituting the critical points into the expressions for length and width obtained in step 5. Finally, calculate the area using these dimensions.

By following these steps, you will be able to answer all the questions related to the given figure.