Question

In the figure, the ratio of AB to BC is 7:5. If AC=1, calculate the distance from A to the midpoint of BC?

(a) $\dfrac{5}{8}$
(b) $\dfrac{2}{3}$
(c) $\dfrac{19}{24}$
(d) $\dfrac{3}{4}$

Answer 1

Hint:Use the basic definition of ratio, to find the values of AB in terms of BC. Then take AB+BC=AC and solve to get the answer.

Complete step-by-step answer:
Let us first know what a ratio is.
A ratio, in basic words, is a quantity used to define a comparison between two quantities. A bit toward the advanced side, it is the quantity which defines how many times of one quantity is that of others.
At our level, apart from the definition, we will treat it as a simple fraction that defines a relation between two given quantities.
Now, starting with the question.

It is given in the question that the ratio of AB to BC is 7:5. This can be mathematically written as:
$\dfrac{AB}{BC}=\dfrac{7}{5}$
$\Rightarrow AB=\dfrac{7}{5}BC..........(i)$
Also, it is given that the length of AB is 1 units.
$\therefore AC=1$
$\Rightarrow AB+BC=1$
Now we will substitute the value of AB from equation (i). On doing so, we get
$\dfrac{7}{5}BC+BC=1$
$\Rightarrow \dfrac{12}{5}BC=1$
$\Rightarrow BC=\dfrac{5}{12}$
Now we are asked to find the distance of A from the midpoint of BC. The distance is given by:
$AB+\dfrac{1}{2}BC=\dfrac{7}{5}BC+\dfrac{1}{2}BC=\dfrac{19}{10}\times \dfrac{5}{12}=\dfrac{19}{24}$
Hence, the correct option is (c), i.e., the distance of point A from midpoint of BC is $\dfrac{19}{24}\text{ units}\text{.}$

Note: Read the question carefully as in the question including ratio, the given ratio we have to take in fraction form and based on the data given substitute the values and find the distance from midpoint of BC to point A.We can also solve this question,by observing the diagram we can calculate distance as difference of AC and midpoint of BC i.e $AC-\dfrac{1}{2}BC$
$=1-\dfrac{1}{2}\times\dfrac{5}{12}$
$=1-\dfrac{5}{24}$
$=\dfrac{19}{24}$
We get the same answer.