Question

The coordinates of point A are (p, q) and coordinates of point B are (p+2q, q+2p). Provide your complete solutions and proofs in your paper homework and respond to questions or statements online. Show that the midpoint of AB is the same distance from the x-axis and y-axis

Answer 1

The mid-point (p + q , q + p) of AB is the same distance from the x-axis and the y-axis

Explanation:

* Lets explain how to solve the problem

- Any point will be equidistant from the x-axis and the y-axis must have

equal coordinates

- Ex: point (4 , 4) is the same distance from the x-axis and the y-axis

because the distance from the x-axis to the point is 4 (y-coordinate)

and the distance from the y-axis and the point is 4 (x-coordinate)

- If (x , y) is the mid-point of a segment its endpoints are

`(x_(1),y_(1))` and
`(x_(2),y_(2))`, then

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

* Lets solve the problem

∵ Point A has coordinates (p , q)

∵ Point B has coordinates (p + 2q , q + 2p)

- The mid-point of AB is (x , y)

∵
`x=(p+p+2q)/(2)`

∴
`x=(2p+2q)/(2)`

- Take 2 as a common factor from the terms of the numerator

∴
`x=(2(p+q))/(2)`

- Divide up and down by 2

∴ x = p + q

∵
`y=(q+q+2p)/(2)`

∴
`y=(2q+2p)/(2)`

- Take 2 as a common factor from the terms of the numerator

∴
`y=(2(q+p))/(2)`

- Divide up and down by 2

∴ y = q + p

∴ The mid point of AB is (p + q , q + p)

- p + q is the same with q + p

∵ The x-coordinate of the mid point of AB is p + q

∵ The y-coordinate of the mid point of AB is q + p

∵ p + q = q + p

∴ The coordinates of the mid-point of AB are equal

- According the explanation above

∴ The mid-point (p + q , q + p) of AB is the same distance from the

x-axis and the y-axis