Find k, given that (2,k) is equidistant from (3,7) and (9,1).
The answer is : k=0
How do I solve this problem?
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Well, first of all, you gotta know that distance is like the spiciest ingredient when it comes to solving this problem. So let's bring in some math flavor, shall we?
The distance between two points (x1, y1) and (x2, y2) can be found using the Pythagorean theorem. It's the same theorem used to figure out how long your favorite triangle's legs are. In this case, our points are (3, 7) and (9, 1).
Now, if (2, k) is the same distance away from both these points, it means the distance from (2, k) to (3, 7) is equal to the distance from (2, k) to (9, 1). Let's break this down:
Distance from (2, k) to (3, 7) = sqrt((3-2)^2 + (7-k)^2)
Distance from (2, k) to (9, 1) = sqrt((9-2)^2 + (1-k)^2)
Since those two distances are equal, we can write an equation:
sqrt((3-2)^2 + (7-k)^2) = sqrt((9-2)^2 + (1-k)^2)
Squaring both sides because we like things spicy and bold, we get:
(3-2)^2 + (7-k)^2 = (9-2)^2 + (1-k)^2
Simplifying this equation, we get:
1 + (7-k)^2 = 49 + (1-k)^2
Expanding and getting rid of those pesky square thingies, we obtain:
1 + 49 - 14k + k^2 = 49 + 1 - 2k + k^2
Simplifying, we end up with:
-14k = -2k
Dividing both sides by -2, we find:
k = 0
Voila! There you have it. The answer to our little puzzle is k = 0. Hope that brought a smile to your mathematical face!
To solve this problem, we need to find the value of k such that the point (2, k) is equidistant from the points (3, 7) and (9, 1).
The distance between two points in a Cartesian plane can be found using the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Let's use this formula to find the distances from (2, k) to (3, 7) and (9, 1):
d1 = sqrt((3 - 2)^2 + (7 - k)^2)
d2 = sqrt((9 - 2)^2 + (1 - k)^2)
Since (2, k) is equidistant from both points, we can set d1 equal to d2 and solve for k:
sqrt((3 - 2)^2 + (7 - k)^2) = sqrt((9 - 2)^2 + (1 - k)^2)
Squaring both sides of the equation, we get:
(3 - 2)^2 + (7 - k)^2 = (9 - 2)^2 + (1 - k)^2
Expanding both sides of the equation, we have:
1 + (7 - k)^2 = 49 + (1 - k)^2
Simplifying further, we get:
1 + 49 - 14k + k^2 = 49 + 1 - 2k + k^2
Combining like terms and canceling out k^2, we have:
-14k = -2k
Dividing both sides of the equation by -2, we get:
k = 0
Therefore, the value of k that satisfies the equation is 0.
?? Samja nai aii
The slope of the line segment joining (3,7) and (9,1) is -6/6 = -1.
Any point equidistant from those two points will lie on the perpendicular bisector of that line segment.
So, we want the line with slope=1, through (6,4), the midpoint of the segment.
y-4 = 1(x-6)
Now, we plug in (2,k) to get
k-4 = 2-6
k = 0