Let O be the origin. Points P and Q lie in the first quadrant. The slope of line segment line OP is 1 and the slope of line segment line OQ is 7. If OP = OQ,$ then compute the slope of line segment line PQ.
Let's first define the point P as (x,x) since the slope of line OP is 1, meaning the y = x. Next, realize that the distance between P to the origin is equal to the distance Q to the origin. The distance can be defined as sqrt(x^2 + x^2) by pythagorean theorem, giving us x(sqrt2) as the distance to the origin. That means that the distance of line segment OQ is x(sqrt2), and since the slope is 7, the point can be represented as (7y, y). With pythagorean theorem, we get 49y^2 + y^2 = 2x^2, which simplifies to 25y^2 = x^2. Square rooting both sides gives us 5y = x. Now, we have the two points P and Q in terms of y. P is equivalent to (5y, 5y), and Q is equivalent to (7y, y). Now we can find the slope of line segment pq with the formula (delta y / delta x), which is just equal to (y-5y) / (7y-5y), simplifing to -4y / 2y, which gives us a slope of -2