How can I solve this problem?

A circle has diameter [AB]. The centre of the circle is (1,3), and B has coordinates (4,-1).

# Find the coordinates of A.
# Find the length of the diameter.

HELP ME PLEASE

A is the same distance from (1,3) as B is.

You don't need to know that the distance is √29. You just need to subtract the same values from (1,3) as were added to get to (4,-1)

(1,3)+(3,-4) = (4,-1)
So, you need
(1,3)-(3,-4) = (-2,7)

diameter= 2*(sqrt((4+1)^2 + (-1+3)^2)

diameter=2 sqrt(25+4)=2sqrt29

Now, A must be on the other side of B, or A is (-2,7)

Why √29 to (-2,7)?

How can I know?

OK, I understand.

Thank you very much!!

To solve this problem, we can use the distance formula and the midpoint formula.

To find the coordinates of point A, we need to find the midpoint of segment [AB]. The midpoint formula states that the coordinates of the midpoint (M) of a segment with endpoints (x1, y1) and (x2, y2) can be found using the following formulas:

x-coordinate of midpoint (M) = (x1 + x2) / 2
y-coordinate of midpoint (M) = (y1 + y2) / 2

Given that point B has coordinates (4,-1) and the center of the circle is (1,3), we can find the coordinates of point A as follows:

x-coordinate of A = 2 * x-coordinate of M - x-coordinate of B
y-coordinate of A = 2 * y-coordinate of M - y-coordinate of B

Now, let's calculate the coordinates of point A:

Step 1: Calculate the x-coordinate of M:
x-coordinate of M = (1 + 4) / 2 = 5 / 2 = 2.5

Step 2: Calculate the y-coordinate of M:
y-coordinate of M = (3 + (-1)) / 2 = 2 / 2 = 1

Step 3: Calculate the x-coordinate of A:
x-coordinate of A = 2 * x-coordinate of M - x-coordinate of B = 2 * 2.5 - 4 = 5 - 4 = 1

Step 4: Calculate the y-coordinate of A:
y-coordinate of A = 2 * y-coordinate of M - y-coordinate of B = 2 * 1 - (-1) = 2 + 1 = 3

So, the coordinates of point A are (1, 3).

To find the length of the diameter of the circle, we can use the distance formula. The distance formula states that the distance between two points (x1, y1) and (x2, y2) can be found using the following formula:

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

Given that the center of the circle is (1, 3) and point B has coordinates (4, -1), we can calculate the length of the diameter as follows:

Distance = √((4 - 1)^2 + (-1 - 3)^2) = √(3^2 + (-4)^2) = √(9 + 16) = √25 = 5

So, the length of the diameter of the circle is 5 units.