Final answer:
By setting up an equation using the fact that Q is the midpoint of PR, and given the segment lengths QR and PR, we solved for x and determined that the measure of PR is 12 units.
Step-by-step explanation:
To find the measure of PR given that Q is the midpoint of PR and the segment lengths are QR = 9 - 3x and PR = 14x - 2, we can set up an equation since the midpoint divides PR into two equal segments. This means that PQ is also equal to 9 - 3x because Q is the midpoint. Therefore, we can express PR as the sum of PQ and QR, giving us:
PR = PQ + QR
Since PQ is equal to QR, we can double the length of QR to get PR:
PR = 2(QR)
Substituting the given value for QR:
PR = 2(9 - 3x)
The equation for PR is:
PR = 14x - 2
By setting the two expressions for PR equal to each other, we can solve for x:
14x - 2 = 2(9 - 3x)
Distribute the 2:
14x - 2 = 18 - 6x
Add 6x to both sides and add 2 to both sides:
20x = 20
Divide both sides by 20 to find x:
x = 1
Now that we know x is 1, we can plug it back into the equation for PR to find the length of PR:
PR = 14x - 2
PR = 14(1) - 2
PR = 14 - 2
PR = 12
So, the measure of PR is 12 units.