Question

in the diagram below of triangle vwx, y is the midpoint of \\( \overline { v x } \\) and z is the midpoint of \\( \overline { w x } \\). if \\( y z = - 7 x + 39 \\), and \\( v w = 48 - 8 x \\), what is the measure of \\( \overline { v w } \\)?

Explanation:

Step1: Use the mid - segment theorem

The mid - segment theorem states that the length of the mid - segment (\(YZ\)) of a triangle is half the length of the parallel side (\(VW\)). So, \(VW = 2YZ\).

Step2: Substitute the given expressions

Given \(YZ=-7x + 39\) and \(VW = 48-8x\). Substitute into \(VW = 2YZ\):
\(48-8x=2(-7x + 39)\)

Step3: Expand the right - hand side

Using the distributive property \(a(b + c)=ab+ac\), we have \(48-8x=-14x + 78\).

Step4: Solve for \(x\)

Add \(14x\) to both sides: \(48-8x + 14x=-14x + 78+14x\), which simplifies to \(48 + 6x=78\).
Subtract \(48\) from both sides: \(6x=78 - 48\), so \(6x=30\).
Divide both sides by \(6\): \(x = 5\).

Step5: Find the length of \(VW\)

Substitute \(x = 5\) into the expression for \(VW\): \(VW=48-8x\).
\(VW=48-8\times5\)
\(VW=48 - 40\)

Answer:

\(8\)