The points P,Q,R and S lie in that order on a straight line . The midpoint of ` bar(QS)` is R and the midpoint of `bar (PS)` is Q . The length of ` bar(QR)` is x feet and the length of ` bar (PQ) ` is `4x-16` feet . What is the length , in feet , of `bar(PS)`?

Options

A. 32
B. 20
C. 16
D. 8

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the relationships between the points
We know that:
- Points P, Q, R, and S lie on a straight line in that order.
- R is the midpoint of segment QS.
- Q is the midpoint of segment PS.
- The length of segment QR is \( x \) feet.
- The length of segment PQ is \( 4x - 16 \) feet.

### Step 2: Establish the lengths of segments based on midpoints
Since R is the midpoint of QS, we have:
- \( QR = x \)
- \( RS = x \)

Thus, the total length of segment QS is:
\[ QS = QR + RS = x + x = 2x \]

Since Q is the midpoint of PS, we have:
- \( PQ = 4x - 16 \)

And since Q is the midpoint of PS:
\[ PQ = QS \]
This gives us the equation:
\[ 4x - 16 = 2x \]

### Step 3: Solve the equation for \( x \)
Now, we will solve the equation:
\[ 4x - 16 = 2x \]

Subtract \( 2x \) from both sides:
\[ 4x - 2x - 16 = 0 \]
\[ 2x - 16 = 0 \]

Add 16 to both sides:
\[ 2x = 16 \]

Now, divide by 2:
\[ x = 8 \]

### Step 4: Find the length of segment PS
Now that we have \( x \), we can find the length of PS. The length of PS can be calculated as:
\[ PS = PQ + QR + RS \]

Substituting the known lengths:
- \( PQ = 4x - 16 \)
- \( QR = x \)
- \( RS = x \)

So,
\[ PS = (4x - 16) + x + x \]
\[ PS = 4x - 16 + 2x \]
\[ PS = 6x - 16 \]

Now substitute \( x = 8 \):
\[ PS = 6(8) - 16 \]
\[ PS = 48 - 16 \]
\[ PS = 32 \]

### Final Answer
The length of segment PS is **32 feet**.

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