Question

w is the midpoint of vx and u is the midpoint of vy. if xy = x - 29 and uw = x - 47, what is the value of x? (there is a diagram with points v, u, y, w, x where u is on vy, w is on vx, and uw and xy are segments)

Explanation:

Step1: Use 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, \(UW\) is the mid - segment of \(\triangle VXY\) (since \(W\) is the midpoint of \(VX\) and \(U\) is the midpoint of \(VY\)). So, \(UW=\frac{1}{2}XY\).

Step2: Substitute the given expressions

Given \(XY = z - 29\) and \(UW=z - 47\). Substitute into the equation \(UW=\frac{1}{2}XY\):
\(z - 47=\frac{1}{2}(z - 29)\)

Step3: Solve the equation for \(z\)

Multiply both sides of the equation by \(2\) to get rid of the fraction:
\(2(z - 47)=z - 29\)
Expand the left - hand side: \(2z-94 = z - 29\)
Subtract \(z\) from both sides: \(2z - z-94=z - z - 29\), which simplifies to \(z-94=-29\)
Add \(94\) to both sides: \(z=-29 + 94\)

Answer:

\(z = 65\)