If R is the midpoint of segment QS, RS = 2x - 4, ST = 4x - 1, and RT = 8x - 43, find QS.
To find the length of segment QS, we need to calculate the sum of the lengths of segments RS and ST.
Given that R is the midpoint of segment QS, we can write the following equation:
RS + ST = QS
Now, let's substitute the given equations for RS, ST, and RT into the equation:
(2x - 4) + (4x - 1) = QS
Simplifying the equation:
6x - 5 = QS
Therefore, the length of segment QS is 6x - 5.
Since R is the midpoint of segment QS, we know that RS = ST. Setting these two expressions equal to each other, we get:
2x - 4 = 4x - 1
Subtracting 2x from both sides, we have:
-4 = 2x - 1
Adding 1 to both sides:
-3 = 2x
Dividing both sides by 2:
x = -3/2
Now we can substitute this value of x back into any of the original expressions to find the lengths of the segments. Let's use RS = 2x - 4:
RS = 2(-3/2) - 4
RS = -3 - 4
RS = -7
Since RS represents half of the length of segment QS, we can multiply -7 by 2 to find QS:
QS = 2 * (-7)
QS = -14
Therefore, the length of segment QS is -14.