Question

q is the midpoint of \\(\overline{pr}\\). if \\(pq = 7x\\) and \\(qr = 2x + 7\\), what is \\(pq\\)? 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

We know \( PQ = 7x \), substitute \( x = \frac{7}{5} \) into it: \( PQ = 7\times\frac{7}{5}=\frac{49}{5}=9.8 \) (or as a fraction \( \frac{49}{5} \) or mixed number \( 9\frac{4}{5} \)). Wait, wait, no, wait. Wait, let's check the equation again. Wait, midpoint means PQ = QR, so 7x = 2x +7. Subtract 2x: 5x=7, so x=7/5. Then PQ is 7x, so 7(7/5)=49/5=9.8? Wait, but maybe I made a mistake. Wait, 7x = 2x +7. 7x -2x =7. 5x=7. x=7/5. Then PQ=7(7/5)=49/5=9.8, which is 9 and 4/5. But let's check again. Alternatively, maybe I messed up the equation. Wait, midpoint: PQ = QR. So 7x = 2x +7. So 5x=7. x=7/5. Then PQ=7*(7/5)=49/5=9.8. Yes, that's correct.

Wait, but let's do it again. Step1: Midpoint implies PQ = QR. So 7x = 2x + 7. Step2: Subtract 2x from both sides: 5x = 7. Step3: Divide by 5: x = 7/5. Step4: PQ = 7x = 7*(7/5) = 49/5 = 9.8, or as a fraction 49/5, or mixed number 9 4/5.

Answer:

\( \frac{49}{5} \) (or 9.8 or \( 9\frac{4}{5} \))