Question

t is the midpoint of \\(\overline{ux}\\) and v is the midpoint of \\(\overline{uw}\\). if \\(wx = y + 7\\) and \\(tv = y - 19\\), what is the value of \\(y\\)?

Explanation:

Step1: Apply the mid - segment theorem

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

Step2: Substitute the given expressions

Given \(WX = y + 7\) and \(TV=y - 19\), substitute into the equation: \(y-19=\frac{1}{2}(y + 7)\).

Step3: Solve the equation

Multiply both sides by 2 to get \(2(y - 19)=y + 7\).
Expand: \(2y-38=y + 7\).
Subtract \(y\) from both sides: \(2y-y-38=y - y+ 7\), so \(y-38 = 7\).
Add 38 to both sides: \(y-38+38=7 + 38\).

Answer:

\(y = 45\)