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.
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.
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).
TextCopy
P R SNow, let’s place Q on this line such that P is to the left of Q, and Q is to the left of S.
TextCopy
P Q R S
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 $QR + RS = QS$
Substituting and Rearranging
From the Segment Addition Postulate, we can write the following equations:
- $PQ + QS = PS$
- $QR + RS = QS$
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.