Question

if $st = w + 69$ and $ru = -2w + 42$, what is the value of $w$? (there is a triangle diagram with points s, t, q, r, u, where r is the midpoint of sq and u is the midpoint of tq, and ru is a midline.)

Explanation:

Step1: Use the mid - segment theorem

The mid - segment theorem states that the length of the mid - segment \(RU\) of a triangle is half the length of the side \(ST\) parallel to it. So, \(RU=\frac{1}{2}ST\).

Step2: Substitute the given expressions

Substitute \(ST = w + 69\) and \(RU=-2w + 42\) into the equation \(RU=\frac{1}{2}ST\). We get \(-2w + 42=\frac{1}{2}(w + 69)\).

Step3: Eliminate the fraction

Multiply both sides of the equation by \(2\) to get \(2(-2w + 42)=w + 69\). Using the distributive property \(a(b + c)=ab+ac\), we have \(-4w+84 = w + 69\).

Step4: Solve for \(w\)

Add \(4w\) to both sides: \(-4w + 4w+84=w + 4w+69\), which simplifies to \(84 = 5w+69\). Then subtract \(69\) from both sides: \(84 - 69=5w+69 - 69\). So, \(15 = 5w\). Divide both sides by \(5\): \(w=\frac{15}{5}=3\).

Answer:

\(3\)