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 = -7x + 44\\), and \\(tu = 9x + 19\\), what is the measure of \\(\overline{tu}\\)?

Explanation:

Step1: Apply Midline Theorem

In triangle \( TUV \), \( W \) is the midpoint of \( \overline{TV} \) and \( X \) is the midpoint of \( \overline{UV} \). By the Midline Theorem (also known as the Midsegment Theorem), the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length. So, \( WX=\frac{1}{2}TU \).

Step2: Set Up the Equation

Substitute \( WX = -7x + 44 \) and \( TU = 9x + 19 \) into the equation from Step 1:

$$ -7x + 44=\frac{1}{2}(9x + 19) $$

Step3: Solve for \( x \)

Multiply both sides of the equation by 2 to eliminate the fraction:

$$ 2(-7x + 44)=9x + 19 $$

$$ -14x + 88 = 9x + 19 $$

Add \( 14x \) to both sides:

$$ 88=23x + 19 $$

Subtract 19 from both sides:

$$ 69 = 23x $$

Divide both sides by 23:

$$ x = 3 $$

Step4: Find the Measure of \( \overline{TU} \)

Substitute \( x = 3 \) into the expression for \( TU \):

$$ TU=9(3)+19 $$

$$ TU = 27 + 19 $$

$$ TU=46 $$

Answer:

The measure of \( \overline{TU} \) is \( \boldsymbol{46} \).