Question

1. if r is the mid - point of $overline{qs}$, $rs = 2x - 4$, $st = 4x - 1$, and $rt = 8x - 43$, find $qs$.

Explanation:

Step1: Use the mid - point property

Since R is the mid - point of QS, then QR = RS. Also, RT=RS + ST. Substitute the given expressions: $8x - 43=(2x - 4)+(4x - 1)$.

Step2: Simplify the right - hand side of the equation

$(2x - 4)+(4x - 1)=2x-4 + 4x-1=6x-5$. So the equation becomes $8x - 43=6x - 5$.

Step3: Solve for x

Subtract 6x from both sides: $8x-6x - 43=6x-6x - 5$, which simplifies to $2x-43=-5$. Then add 43 to both sides: $2x-43 + 43=-5 + 43$, so $2x = 38$. Divide both sides by 2: $x=\frac{38}{2}=19$.

Step4: Find RS

Substitute x = 19 into the expression for RS. RS=$2x - 4=2\times19 - 4=38 - 4 = 34$.

Step5: Find QS

Since R is the mid - point of QS, QS = 2RS. So QS=$2\times34 = 68$.

Answer:

68