A given figure shows a rectangle inscribed in an isoceles right triangle whose hypotenuse is 18 units long. Express the y coordinate of P terms of x

To express the y-coordinate of point P in terms of x, we need to understand the geometry of the problem and use some mathematical principles.

Let's consider the given figure: a rectangle inscribed in an isosceles right triangle whose hypotenuse is 18 units long.

Since the triangle is isosceles, it means that both legs of the triangle (the two sides forming the right angle) are equal in length. Let's call this length "a." Therefore, each leg of the triangle is "a" units long.

The hypotenuse of the triangle is given as 18 units. According to the Pythagorean Theorem, in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Therefore:

a^2 + a^2 = 18^2

Simplifying this equation:

2a^2 = 324

Dividing both sides by 2:

a^2 = 162

Taking the square root of both sides:

a ≈ 12.73

Now that we have the length of the legs, let's proceed to draw the rectangle in the triangle. We place the rectangle such that its sides are parallel to the legs of the triangle. Let's call the length of the rectangle, parallel to the x-axis, "x," and the length perpendicular to the x-axis, "y."

Now, to express the y-coordinate of point P in terms of x, we need to find the relationship between x and y. Looking at the figure, we notice that when we move from point P to the right by x units horizontally, we move upward by y units vertically until we reach the hypotenuse.

Since the hypotenuse is formed by the two legs of the triangle, we can say that moving upward by y units corresponds to moving along the hypotenuse of length "a." We can express this relationship using similar triangles:

y / x = a / a

Simplifying, we find:

y / x = 1

Multiply both sides by x:

y = x

Therefore, the y-coordinate of point P in terms of x is y = x.