Given: Points P,Q,R, and S are collinear
Prove: PQ=PS-WS
To prove that PQ = PS - WS, we need to use the fact that points P, Q, R, and S are collinear.
Let's draw a line segment PR and extend it to point S. So, we have a line segment PQ and a line segment PS, with point R in between them.
Since points P, Q, R, and S are collinear, the distance between P and R plus the distance between R and S should be equal to the distance between P and S.
So, we can say that PR + RS = PS.
Now, we can subtract WS from both sides of the equation:
PR + RS - WS = PS - WS.
Since PQ is equal to PR, we can substitute PR with PQ in the equation:
PQ + RS - WS = PS - WS.
Finally, we can simplify the equation:
PQ + RS - WS = PS - WS.
By subtracting WS from both sides, we have:
PQ + RS = PS.
Therefore, we have proven that PQ = PS - WS using the fact that points P, Q, R, and S are collinear.
To prove that PQ = PS - WS, we first need to make the assumption that PS > WS (PS is greater than WS). We will use this assumption throughout the proof.
Given that points P, Q, R, and S are collinear, we know that they are all on the same line. Let's represent this line as line l.
We can write the equation of line l in slope-intercept form as:
y = mx + b, where m is the slope of line l and b is the y-intercept.
Since points P, Q, and R are collinear, they lie on line l. We can use any two of these points to find the equation of line l.
Let's use points P and Q to find the equation of line l:
P = (x₁, y₁)
Q = (x₂, y₂)
The slope of line l (m) can be calculated using the formula:
m = (y₂ - y₁) / (x₂ - x₁)
Now, let's consider the points P, S, and W.
Since points P, Q, and R are collinear, the slope of line l can also be used to find the coordinates of point S.
Let's assume that the x-coordinate of point S is x₃. Using the slope-intercept form of line l, we can find the y-coordinate of point S using the equation:
y₃ = m * x₃ + b
Now, let's consider the points S and W.
Since we assumed that PS > WS, point W must lie between points P and S. Therefore, the x-coordinate of point W lies between the x-coordinates of points P and S.
Let's assume that the x-coordinate of point W is x₄. Using the slope-intercept form of line l, we can find the y-coordinate of point W using the equation:
y₄ = m * x₄ + b
Now, we can calculate the distances PQ, PS, and WS.
PQ = sqrt((x₂ - x₁)² + (y₂ - y₁)²)
PS = sqrt((x₃ - x₁)² + (y₃ - y₁)²) (Distance formula)
WS = sqrt((x₄ - x₁)² + (y₄ - y₁)²) (Distance formula)
We can substitute the values of y₃ and y₄ using the equations we derived earlier.
PS = sqrt((x₃ - x₁)² + (m * x₃ + b - y₁)²)
WS = sqrt((x₄ - x₁)² + (m * x₄ + b - y₁)²)
Now, let's subtract WS from PS:
PS - WS = sqrt((x₃ - x₁)² + (m * x₃ + b - y₁)²) - sqrt((x₄ - x₁)² + (m * x₄ + b - y₁)²)
At this point, we can see that it is not possible to simplify the expression PS - WS any further without additional information. Therefore, without more specific information about the exact coordinates of points P, Q, R, and S or about the relationship between the distances PS and WS, we cannot prove that PQ = PS - WS.