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Chapter 3: Q17. (page 151)

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

Short Answer

Expert verified

The value of $\frac{A B}{C D} = \frac{3}{5}$ .

Step by step solution

01

– Graphical interpretation of the given situation

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

02

– Find the value of AB:

Given that $A C = \frac{4}{3} A B$ .

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

Substitute $A C = \frac{4}{3} A B$ in $B C = A C - A B$ :

$\begin{matrix} B C = A C 鈭� A B \\ B C = \frac{4}{3} A B 鈭� A B \\ B C = (\frac{4}{3} 鈭� 1) A B \\ B C = \frac{1}{3} A B \\ 3 B C = A B \end{matrix}$

So, $A B = 3 B C .$

03

– Find the value of CD.

Given that $B D = 6 B C$ .

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

Substitute $B D = 6 B C$ in $C D = B D - B C$ :

$\begin{matrix} C D = B D 鈭� B C \\ C D = 6 B C 鈭� B C \\ C D = 5 B C \end{matrix}$

04

– Find the value of ABCD.

Substitute $A B = 3 B C, C D = 5 B C$ in $\frac{A B}{C D}$ :

$\begin{array}{l} \frac{A B}{C D} = \frac{3 B C}{5 B C} \\ \frac{A B}{C D} = \frac{3}{5} \end{array}$

So, the value of $\frac{A B}{C D} = \frac{3}{5}$ .

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