Question

what is the length of $overline{rs}$, when $r$ is the midpoint of $overline{ps}$?

Explanation:

Step1: Set up the equation

Since \(R\) is the mid - point of \(\overline{PS}\), then \(PR = RS\). So we set up the equation \(7x−5 = 4(x + 1)\).

Step2: Expand the right - hand side

Expand \(4(x + 1)\) using the distributive property \(a(b + c)=ab+ac\). We get \(7x−5=4x + 4\).

Step3: Isolate the variable terms

Subtract \(4x\) from both sides of the equation: \(7x-4x−5=4x-4x + 4\), which simplifies to \(3x−5 = 4\).

Step4: Isolate the variable

Add 5 to both sides: \(3x-5 + 5=4 + 5\), so \(3x=9\).

Step5: Solve for \(x\)

Divide both sides by 3: \(\frac{3x}{3}=\frac{9}{3}\), and \(x = 3\).

Step6: Find the length of \(RS\)

Substitute \(x = 3\) into the expression for \(RS\), which is \(4(x + 1)\). So \(RS=4\times(3 + 1)=4\times4 = 16\).

Answer:

16