Question

if u is between t and b, find the value of x and the lengths of the segments. (hint: draw a picture for each problem with the given information and then write the equation to solve.) 11. tu = 2x, ub = 3x + 1, tb = 21 x = tu = ub = 12. tu = 4x - 1, ub = 2x - 1, tb = 5x x = tu = ub = tb =

Explanation:

Step1: Apply segment - addition postulate

Since U is between T and B, then \(TU + UB=TB\). Given \(TU = 2x\), \(UB = 3x + 1\), and \(TB = 21\), we substitute these values into the equation: \(2x+(3x + 1)=21\).

Step2: Simplify the left - hand side

Combine like terms: \(2x+3x+1=(2 + 3)x+1=5x + 1\). So the equation becomes \(5x+1 = 21\).

Step3: Solve for x

Subtract 1 from both sides: \(5x+1−1=21−1\), which gives \(5x=20\). Then divide both sides by 5: \(\frac{5x}{5}=\frac{20}{5}\), so \(x = 4\).

Step4: Find the lengths of TU and UB

For \(TU\), substitute \(x = 4\) into the expression for \(TU\): \(TU=2x=2\times4 = 8\).
For \(UB\), substitute \(x = 4\) into the expression for \(UB\): \(UB=3x + 1=3\times4+1=12 + 1=13\).

Answer:

\(x = 4\), \(TU = 8\), \(UB = 13\)