Let  O be the origin. Points  P and  Q lie in the first quadrant. The slope of line segment OP is  1 and the slope of line segment  is  3. If  then compute the slope of line segment 

Note: The point  (x,y) lies in the first quadrant if both  and  are positive.

 Apr 7, 2024

Understanding the Slopes:

Slope of line segment OP = 1: This means for every 1 unit you move up on the y-axis, you also move 1 unit to the right on the x-axis.


Since O is the origin (0,0), point P will be at (x, x) for some positive value x.


Slope of line segment PQ = 3: This means for every 3 units you move up on the y-axis, you move 1 unit to the right on the x-axis.


Finding Coordinates of P:


As mentioned earlier, P lies at (x, x).


Connecting P and Q:


We know the slope of PQ is 3. We need to find the y-coordinate of point Q relative to P (i.e., how much we move up from P). Let the y-coordinate of Q be y.


Relating Coordinates using Slope:


The difference in y-coordinates (y - x) is 3 times the difference in x-coordinates (which is just x in this case).


Setting up the Equation:


y - x = 3x


Combining Like Terms:


y = 4x


Finding the Slope of PQ:


Slope of PQ = (Change in y) / (Change in x) = (y_Q - y_P) / (x_Q - x_P)


Since both P and Q lie on the positive x-axis (first quadrant), x_Q and x_P are both positive values. Therefore, we can divide both sides of the equation y = 4x by x (assuming x is not zero) to get:


y/x = 4


This gives us the slope of line segment PQ, which is 4.

 Apr 9, 2024

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