Question

Find the coordinates of the point which is three-fourth of the way from A(3,1) to B(2,5).

Answer 1

Hint: In this question, you have to find the point which is three-fourth of the way from A(3,1) to B(2,5). You need to consider a variable for the three-fourth point in the way from A(3,1) to B(2,5). You can use the section formula to solve this problem. So let us see how we can solve this problem.

Complete Step by Step Solution:
Given that there is a three-fourth point in between A(3,1) to B(2,5). Let S be the point which is three-fourth of the way from A(3,1) to B(2,5).
So we have, AP : AB = 3 : 4
Now, AB = AP + PB
 $\therefore \dfrac{{AP}}{{AB}} = \dfrac{{AP}}{{(AP + PB)}} = \dfrac{3}{4}$
On cross multiplying we get,
 $\Rightarrow 4AP = 3AP + 3BP$
Subtracting 3AP from both the sides of the above expression,
 $\Rightarrow 4AP - 3AP = 3BP$
 $\Rightarrow AP = 3BP$
 $\Rightarrow \dfrac{{AP}}{{AB}} = \dfrac{3}{1}$
 So, the ration m : n = 3 : 1
 ${x_1} = 3,{y_1} = 1,{x_2} = 2,{y_2} = 5$
By section formula we get: $x = \dfrac{{m{x_2} + n{x_1}}}{{m + n}}$
Putting all the values of the variable in the above expression we get,
 $\Rightarrow x = \dfrac{{3 \times 2 + 1 \times 3}}{{3 + 1}}$
After addition and subtraction, we get,
 $\Rightarrow x = \dfrac{9}{4}$
 By section formula we get: $y = \dfrac{{m{y_2} + n{y_1}}}{{m + n}}$
 $\Rightarrow y = \dfrac{{3 \times 5 + 1 \times 1}}{{3 + 1}}$
After addition and subtraction, we get,
 $\Rightarrow y = \dfrac{2}{4}$
Therefore, $y = \dfrac{1}{2}$

So the coordinate of P is $(\dfrac{9}{4},\dfrac{1}{2})$.

Note:
In the above solution we have used the section formula. The sectional formula is $M(x,y) = (\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}})$. In the question, it is given that the point is three-fourth from A(3,1) to B(2,5). Section formula is used when we have to find the ratio in which the line segment is divided by a point externally or internally.