given A(a,2) B(3,4) and C(-2,1). if A is at the same distance to the points B and C. Find a
To find the value of 'a' in point A(a,2), we can use the distance formula to determine the distance between point A and points B and C. Since point A is at the same distance from points B and C, we can set up two equations based on the distance formula and solve for 'a'.
The distance between two points (x1, y1) and (x2, y2) is given by the formula:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Let's begin by calculating the distance between points A and B:
d1 = √((3 - a)^2 + (4 - 2)^2)
To simplify the calculation, we can square both sides of the equation to get rid of the square root:
d1^2 = (3 - a)^2 + (4 - 2)^2
Expanding this equation:
d1^2 = (3 - a)^2 + 2^2
d1^2 = (3 - a)^2 + 4
Now, let's find the distance between points A and C:
d2 = √((-2 - a)^2 + (1 - 2)^2)
Squaring both sides:
d2^2 = (-2 - a)^2 + (-1)^2
d2^2 = (-2 - a)^2 + 1
Since we know that the distance from A to B is equal to the distance from A to C, we can set these two equations equal to each other:
(3 - a)^2 + 4 = (-2 - a)^2 + 1
Expanding this equation:
(3 - a)^2 + 4 = (2 + a)^2 + 1
Simplifying and expanding further:
9 - 6a + a^2 + 4 = 4 + 4a + a^2 + 1
Combining like terms:
6 - 6a = 4a + 5
Moving all the variables to one side:
6a + 4a = 5 - 6
10a = -1
Dividing both sides by 10:
a = -1/10
Therefore, the value of 'a' is -1/10.