Geometry question

[gohoyas]

Hello I need help with the following geometry question please:

On a number line, points A,B, and C have coordinates a,b, and c with a<b<c.
If a=9, AC = 21 and AB= 2BC, what are the values of b and c?



Thank you in advance for any help

 

[soroban]

Hello, gohoyas!

Did you make a sketch?

\(\displaystyle \text{On a number line, points }A,B,C\text{ have coordinates }a,b,c\text{ with }a<b<c. \)
\(\displaystyle \text{If }a=9,\;AC = 21,\;AB= 2\!\cdot\!BC,\text{ what are the values of }b\text{ and }c\,?\)

We are given:..\(\displaystyle \begin{array}{ccccc} A && B && C \\ \bullet & ----- & \bullet & --- & \bullet \\ 9 && b && c \end{array}\)


Since \(\displaystyle AC = 21\!:\;c - 9 \:=\:21 \quad\Rightarrow\quad \boxed{c \:=\:30}\)


Then we have:. .\(\displaystyle \begin{array}{ccccc}A && B && C \\ \bullet & ----- & \bullet & --- & \bullet \\ 9 && b && 30 \end{array}\)


Since \(\displaystyle AB = 2\!\cdot\!BC\!:\;b-9 \:=\:2(30-b) \quad\Rightarrow\quad b-9 \:=\:60 - 2b\)

. . . . . . . . . . . . . . . \(\displaystyle 3b \:=\:69 \quad\Rightarrow\quad \boxed{b \:=\:23}\)

 

[gohoyas]

Thank you very much for the detailed explanation..