Question

1. if t is the mid - point of $overline{su}$, find x.
2. if g is the mid - point of $overline{fh}$, find fg.
3. if r is the mid - point of $overline{qs}$, find qs.
4. if b is the mid - point of $overline{ac}$, and $ac = 8x - 20$, find bc.

Explanation:

10.

Step1: Set equal lengths

Since \(T\) is the mid - point of \(\overline{SU}\), then \(ST = TU\). So, \(8x + 11=12x - 1\).

Step2: Solve for \(x\)

Subtract \(8x\) from both sides: \(11 = 4x-1\). Then add 1 to both sides: \(12 = 4x\). Divide both sides by 4, we get \(x = 3\).

Step1: Set equal lengths

Since \(G\) is the mid - point of \(\overline{FH}\), then \(FG=GH\). So, \(11x - 7=3x + 9\).

Step2: Solve for \(x\)

Subtract \(3x\) from both sides: \(8x-7 = 9\). Add 7 to both sides: \(8x=16\). Divide both sides by 8, we get \(x = 2\).

Step3: Find \(FG\)

Substitute \(x = 2\) into the expression for \(FG\): \(FG=11x - 7=11\times2-7=22 - 7 = 15\).

Step1: Set equal lengths

Since \(R\) is the mid - point of \(\overline{QS}\), then \(QR = RS\). So, \(5x-3=21 - x\).

Step2: Solve for \(x\)

Add \(x\) to both sides: \(6x-3 = 21\). Add 3 to both sides: \(6x=24\). Divide both sides by 6, we get \(x = 4\).

Step3: Find \(QS\)

\(QS=QR + RS=(5x - 3)+(21 - x)=4x + 18\). Substitute \(x = 4\) into the expression for \(QS\): \(QS=4\times4+18=16 + 18=34\).

Answer:

\(x = 3\)

11.