Question

in the diagram below of triangle tuv, w is the midpoint of \\( \overline{tv} \\) and x is the midpoint of \\( \overline{uv} \\). if \\( wx = 53 - 6x \\), and \\( tu=-46 + 7x \\), what is the measure of \\( \overline{wx} \\)?

Explanation:

Step1: Use the mid - segment theorem

In a triangle, the mid - segment (a segment connecting the midpoints of two sides) is half the length of the third side. So, \( WX=\frac{1}{2}TU \).

Step2: Substitute the given expressions

Given \( WX = 53-6x \) and \( TU=-46 + 7x \), we substitute into the equation \( WX=\frac{1}{2}TU \).
So, \( 53-6x=\frac{1}{2}(-46 + 7x) \).
Multiply both sides by 2 to get rid of the fraction: \( 2(53-6x)=-46 + 7x \).
Expand the left - hand side: \( 106-12x=-46 + 7x \).

Step3: Solve for \( x \)

Add \( 12x \) to both sides: \( 106=-46 + 7x+12x \).
Simplify the right - hand side: \( 106=-46 + 19x \).
Add 46 to both sides: \( 106 + 46=19x \).
So, \( 152 = 19x \).
Divide both sides by 19: \( x=\frac{152}{19}=8 \).

Step4: Find the length of \( WX \)

Substitute \( x = 8 \) into the expression for \( WX \): \( WX=53-6x \).
\( WX=53-6\times8 \).
\( WX=53 - 48 \).

Answer:

\( 5 \)