Question

Point B has coordinates (1,2). The x-coordinate of point A is a -5. The distance between point A and B is 10 units. What are the possible coordinates of point A?

Answer 1

There are two possible coordinates for A: Either A (-5, 10) or A (-5, -6)

Explanation:

To find the possible coordinates of A, we will need to find the possible y-coordinate.

We can do this using the distance (d) formula, which is

`d=\sqrt{(x_(2)-x_(1))^2+(y_(2)-y_(1))^2 }`, where (x1, x2) are one set of coordinates and (y1, y2) are the other set of coordinates

We can allow point B (1, 2) to be our x1 and x2 coordinates and A (-5, y2) to be our x2 and y2. Thus, we must plug into the formula 10 for d and solve for y2:

`10=\sqrt{(-5-1)^2+(y_(2)-2)^2}\\ 10=\sqrt{(-6)^2+(y_(2)-2)^2}\\ 10=\sqrt{36+(y_(2)-2)^2}\\ 100=36+(y_(2)-2)^2\\64=(y_(2)-2)^2\\`

To finish solving, we must take the square root of both sides. Whenever you take a square root, there is both a positive answer and a negative answer, since squaring a negative number also yields a positive number (e.g., 5 * 5 = 25 and -5 * -5 = 25):

Positive answer:

`8=y_(2)-2\\10=y_(2)`

Negative answer:

`-8=y_(2)-2\\-6=y_(2)`

To check our answers, we can plug in both 10 for y2 into the formula and -6 for y2 into the formula and check that we get 10 each time:

Plugging in 10 for y2:

`10=√((-5-1)^2+(10-2)^2)\\ 10=√((-6)^2+(8)^2)\\ 10=√(36+64)\\ 10=√(100)\\ 10=10`

Plugging in -6 for y2:

`10=√((-5-1)^2+(-6-2)^2)\\ 10=√((-6)^2+(-8)^2)\\ 10=√(36+64)\\ 10=√(100)\\ 10=10`

Thus, the two possible coordinates of A are (-5, 10), where 10 is the y-coordinate or (-5, -6), where - 6 is the y-coordinate.