Question

point u is on line segment \overline{tv}. given uv = 2, tu = 5x, and tv = 3x + 6, determine the numerical length of \overline{tv}.

Explanation:

Step1: Understand the segment addition postulate

Since U is on \( \overline{TV} \), we have \( TU + UV = TV \). Given \( TU = 5x \), \( UV = 2 \) (assuming the "2₁" is a typo and should be 2), and \( TV = 3x + 6 \). So the equation is \( 5x + 2 = 3x + 6 \).

Step2: Solve for x

Subtract \( 3x \) from both sides: \( 5x - 3x + 2 = 3x - 3x + 6 \) → \( 2x + 2 = 6 \). Then subtract 2 from both sides: \( 2x + 2 - 2 = 6 - 2 \) → \( 2x = 4 \). Divide both sides by 2: \( x = \frac{4}{2} = 2 \).

Step3: Find the length of \( \overline{TV} \)

Substitute \( x = 2 \) into \( TV = 3x + 6 \): \( TV = 3(2) + 6 = 6 + 6 = 12 \).

Answer:

12