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 $\left(11,\dfrac{32}{3}\right).$

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

$\left(\dfrac{mc+na}{m+n},\dfrac{md+nb}{m+n}\right).$

Therefore, the co-ordinates of point A will be

$\left(\dfrac{2\times 16+1\times 1}{2+1},\dfrac{2\times 14+1\times 4}{2+1}\right)\\\\\\=\left(\dfrac{32+1}{3},\dfrac{28+4}{3}\right)\\\\\\=\left(\dfrac{33}{3},\dfrac{32}{3}\right)\\\\\\=\left(11,\dfrac{32}{3}\right).$

Thus, the co-ordinates of point A is $\left(11,\dfrac{32}{3}\right).$