Question

Segment PH is graphed on the coordinate grid where P is (-31,8) and H is (29,-10). Point E is the midpoint of segment PH. If point A is graphed 2/3 of the way along the segment from P to E, then what are the coordinates of point A?

Answer 1

The coordinates of point A are (-11 , 2)

Explanation:

- The coordinates of point P are (-31 , 8)

- The coordinates of point H are (29 , -10)

- Point E is the mid-point of segment PH

- The coordinates of the the mid-point are:

→
`x=(x_(1)+x_(2))/(2)` and
`y=(y_(1)+y_(2))/(2)`

∵ E is the mid-point of segment PH

∴ The x-coordinate of point E is
`x=(-31+29)/(2)=-1`

∴ The x-coordinate of point E is -1

∴ The y-coordinate of point E is
`y=(8+-10)/(2)=-1`

∴ The y-coordinate of point E is -1

∴ The coordinates of point E are (-1 , -1)

- Point A is graphed
`(2)/(3)` of the way along the segment from P to E

- That mean the distances from points P to A is 2 parts and from P to E

is 3 parts, then the distance from A to E is "3 - 2" = 1 part

- So point A divides the line segment PE at ratio 2 : 1 from P

- The coordinates of the division point are:

→
`x=(x_(1)m_(2)+x_(2)m_(1))/(m_(1)+m_(2))` and
`y=(y_(1)m_(2)+y_(2)m_(1))/(m_(1)+m_(2))`

∵ Point A divides PE at ratio 2 : 1

∵ The coordinates of point P are (-31 , 8)

∵ The coordinates of point E are (-1 , -1)

∴ The x-coordinate of point A is
`x=((-31)(1)+(-1)(2))/(2+1)= -11`

∴ The x-coordinate of point A is -11

∴ The y-coordinate of point A is
`x=((8)(1)+(-1)(2))/(2+1)=2`

∴ The y-coordinate of point A is 2

∴ The coordinates of point A are (-11 , 2)

* The coordinates of point A are (-11 , 2)