Question

in the diagram below of triangle tuv, w is the midpoint of \\( \overline { t v } \\) and x is the midpoint of \\( \overline { u v } \\). if \\( w x = - 7 x + 44 \\), and \\( t u = 9 x + 19 \\), what is the measure of \\( \overline { t u } \\)?

Explanation:

Step1: Apply the mid - segment theorem

The mid - segment theorem states that the length of the mid - segment (\( WX \)) of a triangle is half the length of the third side (\( TU \)). So, \( TU = 2\times WX \).

Step2: Substitute the given expressions

Given \( WX=-7x + 44 \) and \( TU = 9x+19 \), we substitute into \( TU = 2\times WX \).
We get \( 9x + 19=2(-7x + 44) \).

Step3: Expand the right - hand side

Using the distributive property \( a(b + c)=ab+ac \), \( 2(-7x + 44)=-14x+88 \). So the equation becomes \( 9x + 19=-14x + 88 \).

Step4: Solve for \( x \)

Add \( 14x \) to both sides: \( 9x+14x + 19=-14x+14x + 88 \), which simplifies to \( 23x+19 = 88 \).
Subtract \( 19 \) from both sides: \( 23x+19 - 19=88 - 19 \), so \( 23x=69 \).
Divide both sides by \( 23 \): \( x=\frac{69}{23}=3 \).

Step5: Find the length of \( TU \)

Substitute \( x = 3 \) into \( TU=9x + 19 \).
\( TU=9\times3+19=27 + 19=64 \).

Answer:

\( 64 \)