Answer:
C(3, 2)
D(2.2, 2.8)
Step-by-step explanation:
If point [tex]M(x,y)[/tex] divides the segment AB with endpoints [tex]A(x_1,y_1)[/tex] and [tex]B(x_2,y_2)[/tex] in the ratio [tex]m:n,[/tex] then its coordinates are
[tex]x=\dfrac{n\cdot x_1+m\cdot x_2}{m+n}\\ \\y=\dfrac{n\cdot y_1+m\cdot y_2}{m+n}[/tex]
You are given A(1,4) and B(6,-1).
1. If point C divides AB in the ratio 2 : 3, the coordinates of point C are
[tex]x=\dfrac{3\cdot 1+2\cdot 6}{2+3}=\dfrac{3+12}{5}=\dfrac{15}{5}=3\\ \\y=\dfrac{3\cdot 4+2\cdot (-1)}{2+3}=\dfrac{12-2}{5}=\dfrac{10}{5}=2[/tex]
C(3, 2)
2. If point D divides AC in the ratio 3 : 2, the coordinates of point D are
[tex]x=\dfrac{2\cdot 1+3\cdot 3}{3+2}=\dfrac{2+9}{5}=\dfrac{11}{5}=2.2\\ \\y=\dfrac{2\cdot 4+3\cdot 2}{3+2}=\dfrac{8+6}{5}=\dfrac{14}{5}=2.8[/tex]
D(2.2, 2.8)
Answer:
C is [tex](3,2)[/tex]
D is [tex](2.2,2.8)[/tex]
Step-by-step explanation:
(i) We need to find the coordinate of point C, which divides A(1,4) and B(6,-1) in the ratio 2:3
We know that the coordinate of a point [tex](x,y)[/tex] dividing a line segment joining [tex](x_{1},y_{1}) \:\text{and} (x_{2},y_{2})[/tex] in the ratio m:n is given by
[tex]x=\frac{mx_{2}+nx_{1}}{m+n}[/tex], [tex]y=\frac{my_{2}+ny_{1}}{m+n}[/tex]
Let the coordinate of point C be [tex](x,y)[/tex]
Here, the ratio is 2:3
So, [tex]x=\frac{2(6)+3(1)}{2+3}[/tex]
[tex]x=\frac{15}{5}[/tex]
[tex]x=3[/tex]
[tex]y=\frac{2(-1)+3(4)}{2+3}[/tex]
[tex]y=\frac{-2+12}{5}[/tex]
[tex]y=\frac{10}{5}[/tex]
[tex]y=2[/tex]
Hence, coordinate of C is [tex](3,2)[/tex]
(ii) We need to find the coordinate of point D, which divides A(1,4) and C(3,2) in the ratio 3:2
We know that the coordinate of a point [tex](x,y)[/tex] dividing a line segment joining [tex](x_{1},y_{1}) \:\text{and} (x_{2},y_{2})[/tex] in the ratio m:n is given by
[tex]x=\frac{mx_{2}+nx_{1}}{m+n}[/tex], [tex]y=\frac{my_{2}+ny_{1}}{m+n}[/tex]
Let the coordinate of point D be [tex](x_{3},y_{3})[/tex]
Here, the ratio is 2:3
So, [tex]x_{3}=\frac{3(3)+2(1)}{3+2}[/tex]
[tex]x_{3}=\frac{11}{5}[/tex]
[tex]x_{3}=2.2[/tex]
[tex]y_{3=\frac{3(2)+2(4)}{3+2}[/tex]
[tex]y_{3=\frac{6+8}{5}[/tex]
[tex]y_{3=\frac{14}{5}[/tex]
[tex]y_{3=2.8[/tex]
Hence, coordinate of D is [tex](2.2,2.8)[/tex]
