Question

The endpoints of ab are a(1,4) and b(6,-1). If point c divides ab in the ratio 2:3, the coordinates of c are ?. If point d divides ac in the ratio 3:2, the coordinates of d are ?.

Answer 1

Answer:

• coordinates of c are: (3,2)

• coordinates of d are: (2.2,2.8)

Step-by-step explanation:

• We are given that the  endpoints of ab are a(1,4) and b(6,-1).

If point c divides ab in the ratio 2:3.

We know that if any point C(i,j) divide the directed line segment AB( A(e,f) and B (g,h) ) in the ration m:n,

Then the coordinates of this new point C is given by:

$i=\dfrac{mg+ne}{m+n}$ and $j=\dfrac{mh+nf}{m+n}$

Here we have m:n=2:3.

C=C, A=a, B=b

Hence the coordinates of C is given by:

$i=\dfrac{2\times 6+3\times 1}{2+3}=\dfrac{12+3}{5}=3$

and $j=\dfrac{2\times -1+3\times 4}{2+3}=\dfrac{-2+12}{5}=\dfrac{10}{5}=2$

Hence, the coordinates of c are: (3,2)

• Now we are given that point d divides ac in the ratio 3:2

Hence, here we have A=a, B=c and C=d

Hence the coordinates of d are:

$i=\dfrac{3\times 3+2\times 1}{3+2}=\dfrac{9+2}{5}=\dfrac{11}{5}=2.2$

and $j=\dfrac{3\times 2+2\times 4}{3+2}=\dfrac{6+8}{5}=\dfrac{14}{5}=2.8$

Hence, the coordinates of d are: (2.2,2.8)

Answer 2

Answer:

The coordinate C:(3,2) and coordinate $D:\left ( \dfrac{11}{5},\dfrac{14}{5}, \right )$

Step-by-step explanation:

1. The endpoints of AB are A(1,4) and B(6,-1)

If point C divide AB in the ratio 2:3

Formula:

$\text{Section formula}(x,y)=\left ( \dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+ny_1}{m+n}, \right )$

Where,

$(x_1,y_1)\rightarrow A(1,4)$

$(x_2,y_2)\rightarrow B(6,-1)$

$m:n\rightarrow 2:3$

$C:(x,y)$

$C:\left ( \dfrac{2\cdot 6+3\cdot 1}{2+3},\dfrac{2\cdot -1+3\cdot 4}{2+3}, \right )$

$C:\left ( \dfrac{12+3}{5},\dfrac{-2+12}{5}, \right )$

$C:(3,2)$

2. The endpoints of AC are A(1,4) and B(3,2)

If point D divide AC in the ratio 3:2

$(x_1,y_1)\rightarrow A(1,4)$

$(x_2,y_2)\rightarrow C(3,2)$

$m:n\rightarrow 3:2$

$D:(x,y)$

$D:\left ( \dfrac{3\cdot 3+2\cdot 1}{2+3},\dfrac{3\cdot -1+2\cdot 4}{2+3}, \right )$

$D:\left ( \dfrac{9+2}{5},\dfrac{6+8}{5}, \right )$

$D:\left ( \dfrac{11}{5},\dfrac{14}{5}, \right )$

Hence, The coordinate C:(3,2) and coordinate D:$\left ( \dfrac{11}{5},\dfrac{14}{5}, \right )$