The accompanying figure shows a rectangle inscribed in an isosceles right triangle whose hypotenuse is 

10

units long.

**a.** Express the​ y-coordinate of P in terms of x.​ (You might start by writing an equation for the line​ AB.)

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.

Learn how to find the y-coordinate of point P in an isosceles right triangle inscribed with a rectangle. This problem involves understanding the geometry of isosceles triangles and using the equation of a line to derive the y-coordinate. Suitable for high school students studying coordinate geometry and triangle properties.

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