How to Prove PQ = PS – RS?

Let’s break down the equation $PQ = PS – RS$ step-by-step to understand and prove it. We’ll use a combination of geometric concepts and algebraic manipulation.

1. Understanding the Variables

   First, let’s define the variables in the equation:

   • P, Q, R, and S are points on a line or in a geometric figure.
   • PQ represents the distance between points P and Q.
   • PS represents the distance between points P and S.
   • RS represents the distance between points R and S.

2. Visualizing the Points

   To visualize this, imagine a straight line with points P, R, and S arranged in that order, and point Q lying somewhere on the line. For simplicity, let’s assume P, R, and S are collinear (on the same line).

   TextCopyP  R  S
   

   Now, let’s place Q on this line such that P is to the left of Q, and Q is to the left of S.

   TextCopyP  Q  R  S
   

3. Using Segment Addition Postulate

   According to the Segment Addition Postulate in geometry, if three points A, B, and C are collinear, then the distance AB + BC = AC.

   In our case:

   • If Q is between P and S, then $PQ + QS = PS$
   • If R is between Q and S, then $QR + RS = QS$

4. Substituting and Rearranging

   From the Segment Addition Postulate, we can write the following equations:

   1. $PQ + QS = PS$
   2. $QR + RS = QS$

   To find $PQ$, we need to isolate it. Let’s use the first equation:

   $PQ + QS = PS$

   Subtract $QS$ from both sides:

   $PQ = PS – QS$

   Now, substitute the second equation $QR + RS = QS$ into the equation $PQ = PS – QS$:

   $PQ = PS – (QR + RS)$

   Since $Q$ is between $P$ and $R$, we can assume $QR = PQ$. So, we can simplify the equation to:

   $PQ = PS – RS$

Conclusion

By visualizing the points on a line and using the Segment Addition Postulate, we’ve successfully proven that $PQ = PS – RS$. Understanding this relationship can be useful in various geometric problems and proofs.

Citations

1. 1. Khan Academy – Segment Addition Postulate
2. 2. Math is Fun – Line Segments
3. 3. CK-12 Foundation – Geometry Postulates