Find the possible values of a, if the distance between the points is 5, and the coordinates are (1,1) and (4,a)
I'm not quite sure how to do this one.. I've tried the distance and midpoint formula, but I'm confused.
I don't what to do with the (a-1)^2 part.
5= (4-0)^2 + (a-1)^2
5= 16 + (a-1)^2
Any help or can give me a site with detailed examples would greatly be appreciated. Thanks.
The formula for distance between two point is
d = √( (y2-y1)^2 + (x2-x1)^2
so
5 = √((a-1)^2 ++ (4-1)^2 )
square both sides
25 = (a-1)^2 = 9
16 = (a-1)^2
take √ of both sides
4 = ± (a-1)
a-1 = ±4
a = 5 or a= -3
Thank you Reiny!
To find the possible values of a in this problem, you can use the distance formula. The distance formula is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, you are given two points: (1,1) and (4,a). The x-coordinate of the first point is 1, the y-coordinate of the first point is also 1. The x-coordinate of the second point is 4, and the y-coordinate of the second point is a.
Plugging these values into the distance formula, we get:
5 = sqrt((4 - 1)^2 + (a - 1)^2)
Simplifying:
25 = (4 - 1)^2 + (a - 1)^2
25 = 9 + (a - 1)^2
Subtracting 9 from both sides:
16 = (a - 1)^2
Now, to find the possible values of a, we take the square root of both sides:
±4 = a - 1
So, a could be either 5 or -3. Therefore, the possible values of a are 5 and -3.
You can also solve this problem by considering the midpoint formula. The midpoint formula is given by:
((x1 + x2) / 2, (y1 + y2) / 2)
In this case, the midpoint is ((1 + 4) / 2, (1 + a) / 2), which simplifies to (2.5, (1 + a) / 2).
Since the distance between the two points is 5, the distance from the midpoint to either of the points is also 5. Using the distance formula again, we get:
5 = sqrt((4 - 2.5)^2 + (a - (1 + a) / 2)^2)
Simplifying:
5 = sqrt((1.5)^2 + ((a - 1) / 2)^2)
25 = (1.5)^2 + ((a - 1) / 2)^2
25 = 2.25 + ((a - 1) / 2)^2
Subtracting 2.25 from both sides:
22.75 = ((a - 1) / 2)^2
Now, taking the square root of both sides:
±sqrt(22.75) = (a - 1) / 2
Multiplying both sides by 2:
±2sqrt(22.75) = a - 1
Adding 1 to both sides:
±2sqrt(22.75) + 1 = a
So, a could be either 1 + 2sqrt(22.75) or 1 - 2sqrt(22.75). Therefore, the possible values of a are 1 + 2sqrt(22.75) and 1 - 2sqrt(22.75).
I hope this explanation helps you understand how to solve this problem. If you need more examples or further clarification, feel free to ask!