Question

q is the midpoint of \\(\overline{pr}\\). if \\(pq = 7x\\) and \\(qr = 2x + 7\\), what is \\(pq\\)? diagram of p---q---r simplify your answer and write it as a proper fraction, mixed number, or integer.

Explanation:

Step1: Use midpoint property

Since Q is the midpoint of \( \overline{PR} \), \( PQ = QR \). So we set up the equation \( 7x = 2x + 7 \).

Step2: Solve for x

Subtract \( 2x \) from both sides: \( 7x - 2x = 2x + 7 - 2x \), which simplifies to \( 5x = 7 \). Then divide both sides by 5: \( x = \frac{7}{5} \).

Step3: Find PQ

Now substitute \( x = \frac{7}{5} \) into \( PQ = 7x \). So \( PQ = 7\times\frac{7}{5} = \frac{49}{5} = 9\frac{4}{5} \)? Wait, no, wait. Wait, let's check again. Wait, 7x = 2x +7. Subtract 2x: 5x=7, so x=7/5. Then PQ is 7x, so 7(7/5)=49/5? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, 7x = 2x +7. 7x -2x =7. 5x=7. x=7/5. Then PQ is 7(7/5)=49/5? But 49/5 is 9.8. Wait, but maybe I messed up. Wait, no, let's do it again. Midpoint means PQ=QR. So 7x = 2x +7. Subtract 2x: 5x=7. x=7/5. Then PQ=7x=7(7/5)=49/5? Wait, but 49/5 is 9.8, which is 9 and 4/5. But maybe I made a mistake in the equation. Wait, the problem says Q is the midpoint, so PQ=QR. So 7x=2x+7. Yes, that's correct. So solving for x: 5x=7, x=7/5. Then PQ=7(7/5)=49/5. Wait, but 49/5 is 9.8, which is 9 4/5. But let me check again. Wait, 7x=2x+7. 7x-2x=7. 5x=7. x=7/5. Then PQ=7*(7/5)=49/5. Yes, that's correct.

Wait, but maybe I made a mistake in the problem. Wait, the problem is PQ=7x and QR=2x+7. So midpoint, so PQ=QR. So 7x=2x+7. So 5x=7. x=7/5. Then PQ=7*(7/5)=49/5. So 49/5 is 9.8, which is 9 4/5. But let me confirm. Alternatively, maybe the problem is written wrong? No, the user provided the problem. So according to the steps, that's the solution.

Wait, wait, no, wait a second. 7x = 2x +7. 7x -2x =7. 5x=7. x=7/5. Then PQ=7x=7*(7/5)=49/5. So 49/5 is 9.8, which is 9 and 4/5. So that's the answer.

Answer:

\( \frac{49}{5} \) (or \( 9\frac{4}{5} \) or 9.8, but as a fraction, \( \frac{49}{5} \))