Equation of a Line in Isosceles Triangle

[Jan Hill]

Homework Statement

Given an right angled isosceles triangle with hypoteneuse of 2 units. The hypoteneuse lies on the x-axis and is 2 units long. The points of the triangle are in terms of x and y as follows: C-1,0), A(1,0)and B(0,y). Inscribed in the triangle is a rectangle which has one point on the line AB of the triangle at point P(x, ?) I have to express the y co-ordinate of P in terms of x

Homework Equations

The Attempt at a Solution

I would like to find the equation of the line AB but I don't have the slope. I only have the one point A(1,0) but I think the isosceles nature of the triangle is the key to this. I don't know how that info though.

 

[Tac-Tics]

I didn't quite understand what you meant past the description of the triangle, but it looks like you just need to solve for the coordinate of the point B to help you figure it out.

First, let O be the origin (0, 0).

What is the distance of the line segment AB? (Hint, use the pythagorean theorem and the fact that the length of AB equal the length of BC).

Note that BOA defines another right triangle. Assuming you know the length of AB and the length of OA, you can use the pythagorean theorem again to find OB (and thus, the y-coordinate of B).

Hope that helps.

 

[HallsofIvy]

The line from A to B and the line from C to B have the same length: x. Then, by the Pythagorean theorem, $x^2+ x^2= 2x^2= 4$ so that $x= \sqrt{2}$. Now you know that the line from (0, 0) to (0, y) is a leg in a right triangle with the other leg of length 1 ((0,0) to A) and hypotenuse of length $\sqrt{2}$. The Pythagorean theorem, again, \(\displaystyle y^2+ 1= 2\). That gives you point B.