Question

s is the midpoint of \\(\overline{rt}\\) and v is the midpoint of \\(\overline{rw}\\). if \\(wu = 15w - 15\\) and \\(sv = 3w + 24\\), what is the value of \\(w\\)?

Explanation:

Step1: Apply the mid - segment theorem

The mid - segment theorem states that the length of the mid - segment (a segment connecting the midpoints of two sides of a triangle) is half the length of the third side. Here, \(SV\) is the mid - segment of \(\triangle RTU\) (since \(S\) is the midpoint of \(RT\) and \(V\) is the midpoint of \(RU\)), so \(SV=\frac{1}{2}TU\).

Step2: Substitute the given expressions

Given \(TU = 15x-15\) and \(SV=3x + 24\). Substitute into the equation \(SV=\frac{1}{2}TU\):
\(3x + 24=\frac{1}{2}(15x-15)\)

Step3: Solve the equation

Multiply both sides of the equation by \(2\) to get rid of the fraction:
\(2(3x + 24)=15x-15\)
\(6x+48 = 15x-15\)
Subtract \(6x\) from both sides:
\(48=15x - 6x-15\)
\(48 = 9x-15\)
Add \(15\) to both sides:
\(48 + 15=9x\)
\(63=9x\)
Divide both sides by \(9\):
\(x=\frac{63}{9}=7\)

Answer:

\(7\)