How can you determine the length of segment qr if point m is the midpoint, given that qm is represented as 2x + 5 and mr is represented as 5x + 1?

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To find the length of segment qr, we first need to understand that since point m is the midpoint of segment qr, the lengths of segments qm and mr must be equal. This gives us the equation:

  • qm = mr

Given:

  • qm = 2x + 5
  • mr = 5x + 1

Now we can set up the equation:

2x + 5 = 5x + 1

Next, we solve for x:

  1. Subtract 2x from both sides:
    2. 5 = 3x + 1
  2. Subtract 1 from both sides:
    3. 4 = 3x
  3. Divide both sides by 3:
    4. x = \frac{4}{3}

Now that we have the value of x, we can substitute it back into the expressions for qm and mr to find their lengths:

qm = 2x + 5 = 2\left(\frac{4}{3}\right) + 5 = \frac{8}{3} + \frac{15}{3} = \frac{23}{3}
mr = 5x + 1 = 5\left(\frac{4}{3}\right) + 1 = \frac{20}{3} + \frac{3}{3} = \frac{23}{3}

Since both lengths are equal, we confirm that:

qm = mr = \frac{23}{3}

Finally, the length of the segment qr is the sum of qm and mr:

qr = qm + mr = \frac{23}{3} + \frac{23}{3} = \frac{46}{3}

Thus, the length of segment qr is:

\frac{46}{3} \text{ units}

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