Question

4-1 & 4-2 intro to segments and angles
is point q the midpoint of \\(\overline{pr}\\)?
diagram: p---q---r, pq labeled 5(x + 2), qr labeled 8x - 3, pr total 72
\\(\square\\) point q \\(\square\\) the midpoint of \\(\overline{pr}\\) because x = \\(\square\\), pq = \\(\square\\), and qr = \\(\square\\)

Explanation:

Step1: Set PQ = QR (midpoint definition)

If Q is the midpoint of \( \overline{PR} \), then \( PQ = QR \). So \( 5(x + 2)=8x - 3 \).

Step2: Solve for x

Expand left side: \( 5x + 10 = 8x - 3 \).
Subtract \( 5x \) from both sides: \( 10 = 3x - 3 \).
Add 3 to both sides: \( 13 = 3x \)? Wait, no, wait. Wait, total length \( PR = PQ + QR = 72 \). So actually, \( 5(x + 2)+(8x - 3)=72 \). Let's correct that.
So \( 5x + 10 + 8x - 3 = 72 \).
Combine like terms: \( 13x + 7 = 72 \).
Subtract 7: \( 13x = 65 \).
Divide by 13: \( x = 5 \).

Step3: Find PQ and QR

\( PQ = 5(x + 2)=5(5 + 2)=5\times7 = 35 \).
\( QR = 8x - 3 = 8\times5 - 3 = 40 - 3 = 37 \). Wait, but 35 ≠ 37. Wait, maybe the first approach was wrong. Wait, the total length is 72, so \( PQ + QR = 72 \). So \( 5(x + 2)+(8x - 3)=72 \). Let's redo:
\( 5x + 10 + 8x - 3 = 72 \)
\( 13x + 7 = 72 \)
\( 13x = 65 \)
\( x = 5 \). Then \( PQ = 5(5 + 2)=35 \), \( QR = 8(5)-3 = 37 \). But 35 ≠ 37, so Q is not the midpoint? Wait, maybe the problem is to check if Q is midpoint, so we need to see if PQ = QR. Let's set \( 5(x + 2)=8x - 3 \):
\( 5x + 10 = 8x - 3 \)
\( 10 + 3 = 8x - 5x \)
\( 13 = 3x \) → \( x = 13/3 ≈ 4.333 \), but then total length would be \( 5(13/3 + 2)+(8*(13/3)-3)=5(19/3)+(104/3 - 9/3)=95/3 + 95/3 = 190/3 ≈ 63.333 ≠ 72 \). So the correct approach is total length is 72, so sum of PQ and QR is 72. So:
\( 5(x + 2) + (8x - 3) = 72 \)
\( 5x + 10 + 8x - 3 = 72 \)
\( 13x + 7 = 72 \)
\( 13x = 65 \)
\( x = 5 \)
Then PQ = 5(5+2)=35, QR=8*5 -3=37. Since 35≠37, Q is not the midpoint. Wait, but the problem has dropdowns: first dropdown: "is" or "is not", second dropdown: "is" or "is not" (the midpoint). Wait, let's re-express:

If Q is the midpoint, then PQ = QR and PQ + QR = PR = 72, so PQ = QR = 36. Let's check if with x, PQ and QR can be 36.

For PQ: 5(x + 2)=36 → x + 2 = 36/5 = 7.2 → x=5.2

For QR: 8x - 3=36 → 8x=39 → x=39/8=4.875. Different x, so not midpoint.

But according to the total length, x=5, PQ=35, QR=37. So Q is not the midpoint.

Wait, maybe the problem is to fill in:

First dropdown: "is not"

Second dropdown: "is not"

x=5

PQ=35

QR=37

Let's confirm the calculation again:

Total length PR = PQ + QR = 5(x+2) + (8x - 3) = 5x +10 +8x -3 =13x +7. Given PR=72, so 13x +7=72 →13x=65→x=5. Then PQ=5(5+2)=35, QR=8*5 -3=37. Since 35≠37, Q is not the midpoint.

Answer:

First dropdown: is not; Second dropdown: is not; \( x = 5 \); \( PQ = 35 \); \( QR = 37 \)