Theory and Examples


The accompanying figure shows a rectangle inscribed in an isosceles right triangle whose hypotenuse is 2 units long.


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


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Verified step by step guidance

1. Identify the properties of the isosceles right triangle: Since the hypotenuse is 2 units long, each leg of the triangle is \( \frac{2}{\sqrt{2}} = \sqrt{2} \) units long.
2. Determine the equation of the line AB: The line AB is the hypotenuse of the triangle, which can be expressed as \( y = -x + \sqrt{2} \) because it has a negative slope and passes through the point (\( \sqrt{2}, 0 \)).
3. Consider the rectangle inscribed in the triangle: The top right corner of the rectangle, point P, lies on the line AB.
4. Express the y-coordinate of P in terms of x: Since point P lies on the line AB, its y-coordinate can be expressed using the line's equation as \( y = -x + \sqrt{2} \).
5. Conclude the expression: Therefore, the y-coordinate of point P in terms of x is \( y = -x + \sqrt{2} \).

Key Concepts

Isosceles Right Triangle Properties

An isosceles right triangle has two equal sides and a right angle, with the hypotenuse opposite the right angle. In this case, the hypotenuse measures 2 units, which allows us to determine the lengths of the legs using the Pythagorean theorem. Each leg will be √2 units long, providing a basis for further calculations involving the inscribed rectangle.

Equation of a Line

The equation of a line can be expressed in the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. For the line AB in the triangle, identifying the coordinates of points A and B will allow us to calculate the slope and subsequently derive the equation that relates x and y coordinates, which is essential for expressing the y-coordinate of point P.

Inscribed Figures

An inscribed figure is one that is contained within another shape, touching it at certain points. In this scenario, the rectangle is inscribed within the isosceles right triangle, meaning its vertices lie on the triangle's sides. Understanding the relationship between the dimensions of the inscribed rectangle and the triangle's geometry is crucial for deriving the necessary equations and relationships.