Question

1. if r is the mid - point of $overline{qs}$, $rs = 2x - 4$, $st = 4x - 1$, and $rt = 8x - 49$, find qs.

Explanation:

Step1: Use the mid - point property

Since \(R\) is the mid - point of \(QS\), \(QR = RS\). Also, \(RT=RS + ST\). Substitute \(RS = 2x - 4\), \(ST=4x - 1\), and \(RT = 8x-49\) into the equation \(RT=RS + ST\).
\(8x-49=(2x - 4)+(4x - 1)\)

Step2: Simplify the right - hand side of the equation

\((2x - 4)+(4x - 1)=2x-4 + 4x-1=6x-5\). So the equation becomes \(8x-49 = 6x-5\).

Step3: Solve for \(x\)

Subtract \(6x\) from both sides: \(8x-6x-49=6x-6x - 5\), which simplifies to \(2x-49=-5\). Then add 49 to both sides: \(2x-49 + 49=-5 + 49\), so \(2x = 44\). Divide both sides by 2: \(x = 22\).

Step4: Find \(RS\)

Substitute \(x = 22\) into the expression for \(RS\): \(RS=2x - 4=2\times22-4=44 - 4=40\).

Step5: Find \(QS\)

Since \(R\) is the mid - point of \(QS\), \(QS = 2RS\). So \(QS=2\times40 = 80\).

Answer:

80