Question

The endpoints of line MP are M(1,4) and P(16,14). If A partitions line MP in a ratio of MA: AP=2: 1, which of the following represents the coordinates of point A?

A(6,9)
B(7,8)
C(11,32/3)
D(9,6)

Answer 1

we have that
point M(1,4)
and
point P(16,14)

the distance MP
in the x coordinate is
d MPx=(16-1)-----> 15 units

the distance MP
in the y coordinate is
d MPy=(14-4)-----> 10 units

the coordinate of point A
in the x coordinate
Mx=1----------> the x coordinate of the point M
Ax=Mx+(2/3)*d MPx-------> 1+(2/3)*15------> Ax=11

the coordinate of point A
in the y coordinate
My=4----------> the y coordinate of the point M
Ay=My+(2/3)*d MPy-------> 4+(2/3)*10------> (12+20)/3------> Ay=32/3

the coordinates of point A is (11,32/3)

the answer is the option
C(11,32/3)

Answer 2

Answer: The co-ordinates of point A is
`\left(11,(32)/(3)\right).`

Step-by-step explanation: Given that the co-ordinates of the endpoints of line MP are M(1,4) and P(16,14) and a point A partitions line MP in a ratio of MA: AP=2: 1.

We are to select the correct co-ordinates of the point A.

The co-ordinates of a point that divides the line segment with co-ordinates of the end-points (a, b) and (c, d) in the ratio m : n internally is given by

`\left((mc+na)/(m+n),(md+nb)/(m+n)\right).`

Therefore, the co-ordinates of point A will be

`\left((2* 16+1* 1)/(2+1),(2* 14+1* 4)/(2+1)\right)\\\\\\=\left((32+1)/(3),(28+4)/(3)\right)\\\\\\=\left((33)/(3),(32)/(3)\right)\\\\\\=\left(11,(32)/(3)\right).`

Thus, the co-ordinates of point A is
`\left(11,(32)/(3)\right).`