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, we have $TU + UB=TB$. 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 of the equation

Combine like terms: $2x+3x+1 = 5x + 1$. 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 length of TU and UB

Substitute $x = 4$ into the expressions for TU and UB. $TU=2x=2\times4 = 8$, $UB=3x + 1=3\times4+1=12 + 1=13$.

Step1: Apply segment - addition postulate

Since U is between T and B, $TU+UB = TB$. Given $TU = 4x-1$, $UB=2x - 1$, and $TB = 5x$, we substitute these values into the equation: $(4x-1)+(2x - 1)=5x$.

Step2: Simplify the left - hand side of the equation

Combine like terms: $4x-1+2x - 1=6x-2$. So the equation becomes $6x-2 = 5x$.

Step3: Solve for x

Subtract $5x$ from both sides: $6x-5x-2=5x-5x$, which gives $x-2 = 0$. Then add 2 to both sides: $x-2 + 2=0 + 2$, so $x = 2$.

Step4: Find the length of TU, UB, and TB

Substitute $x = 2$ into the expressions. $TU=4x-1=4\times2-1=8 - 1=7$, $UB=2x - 1=2\times2-1=4 - 1=3$, $TB=5x=5\times2 = 10$.

Answer:

$x = 4$, $TU = 8$, $UB = 13$

Now for problem 12: