A line segment begins at point (3,4) and has a length of 13 units. What are the possible endpoints of this line segment?
(x-3)^2+(y-4)^2=169
that equation describes a circle of radius 13 around 3,4
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To find the possible endpoints of the line segment, we need to consider all points that are 13 units away from (3,4).
The distance between two points, (x1, y1) and (x2, y2), is given by the distance formula:
d = √[(x2 - x1)^2 + (y2 - y1)^2]
In this case, (x1, y1) = (3,4). Let the endpoints be (x, y). Plugging these values into the distance formula, we get:
13 = √[(x - 3)^2 + (y - 4)^2]
Squaring both sides of the equation, we get:
169 = (x - 3)^2 + (y - 4)^2
Expanding and simplifying, we have:
169 = x^2 - 6x + 9 + y^2 - 8y + 16
Rearranging the terms, we get:
x^2 - 6x + y^2 - 8y + 144 = 0
Now, we need to find the values of x and y that satisfy this equation.
There are multiple ways to proceed from here, but one method is to complete the square for both the x and y terms.
Completing the square for the x terms, we have:
x^2 - 6x = -(y^2 - 8y + 144)
To complete the square for the x terms, we need to take half of the coefficient of x, which is -6, square it, and add it to both sides:
x^2 - 6x + (-6/2)^2 = -(y^2 - 8y + 144) + (-6/2)^2
Simplifying, we get:
x^2 - 6x + 9 = -(y^2 - 8y + 144) + 9
x^2 - 6x + 9 = -(y^2 - 8y + 153)
Completing the square for the y terms follows a similar process:
y^2 - 8y = -(x^2 - 6x + 9) + 153
To complete the square for the y terms, we need to take half of the coefficient of y, which is -8, square it, and add it to both sides:
y^2 - 8y + (-8/2)^2 = -(x^2 - 6x + 9) + 153 + (-8/2)^2
Simplifying, we get:
y^2 - 8y + 16 = -(x^2 - 6x + 9) + 153 + 16
y^2 - 8y + 16 = -(x^2 - 6x + 25) + 169
At this point, we can rewrite the equation as:
(x - 3)^2 + (y - 4)^2 = 13^2
Now, we recognize that this equation represents a circle with its center at (3,4) and a radius of 13 units. Therefore, any point on the circumference of this circle will be a possible endpoint of the line segment.
In conclusion, the possible endpoints of the line segment are all the points on the circumference of the circle centered at (3,4) with a radius of 13 units.