Question

Points$A,B,C$, and $D$lie in consecutive order on a line. If $AC=\frac{4}{3}AB$and $BD=6BC$, then what is$\frac{AB}{CD}$?

Short answer

The value of $\frac{AB}{CD}=\frac{3}{5}$.

Step-by-step solution

– Graphical interpretation of the given situation

The points $A,B,C$and D are lie on the line in a consecutive order.

– Find the value of AB:

Given that $AC=\frac{4}{3}AB$.

From the diagram notice that, the length of $BC$is obtained by subtracting the length of AB from AC. That is, $BC=AC-AB$.

Substitute $AC=\frac{4}{3}AB$in $BC=AC-AB$:

$\begin{array}{c}BC=AC-AB\\ BC=\frac{4}{3}AB-AB\\ BC=\left(\frac{4}{3}-1\right)AB\\ BC=\frac{1}{3}AB\\ 3BC=AB\end{array}$

So, $AB=3BC.$

– Find the value of CD.

Given that $BD=6BC$.

From the diagram notice that, the length of CD is obtained by subtracting the length of BC from BD. That is, $CD=BD-BC$.

Substitute $BD=6BC$in $CD=BD-BC$:

$\begin{array}{c}CD=BD-BC\\ CD=6BC-BC\\ CD=5BC\end{array}$

– Find the value of ABCD.

Substitute $AB=3BC,CD=5BC$in $\frac{AB}{CD}$:

$\begin{array}{l}\frac{AB}{CD}=\frac{3BC}{5BC}\\ \frac{AB}{CD}=\frac{3}{5}\end{array}$

So, the value of $\frac{AB}{CD}=\frac{3}{5}$.