find the point on the directed segment from (-3,4) to (6,1) that divides it into a ratio of 1:2.
the ratio divides the segment into pieces of size 1/3 and 2/3.
1/3 of the way from (-3,4) to (6,1) is (3,-1)
(-3,4)+(3,-1) = (0,3)
Thanks
Write the equation of the line that has a slope of 3 and a y-intercept of -8
To find the point that divides the directed segment from (-3,4) to (6,1) into a ratio of 1:2, we will use the concept of a scalar.
1. First, calculate the distance of the entire segment. The distance between two points (x1, y1) and (x2, y2) can be found using the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, using the given points (-3,4) and (6,1), the distance of the segment is:
Distance = sqrt((6 - (-3))^2 + (1 - 4)^2) = sqrt(9^2 + (-3)^2) = sqrt(81 + 9) = sqrt(90)
2. Now, let's assume the point dividing the segment in a ratio of 1:2 as P. We can denote the coordinates of P as (x, y).
3. Since the segment is divided in a ratio of 1:2, it means that the distance from (-3,4) to P is one part and the distance from P to (6,1) is two parts.
Using the ratio, we can write the following equation:
Distance from (-3,4) to P / Distance from P to (6,1) = 1 / 2
Now, substitute the distance values we calculated earlier:
(sqrt(90) - Distance from (-3,4) to P) / (Distance from P to (6,1)) = 1 / 2
4. Solve the equation to find the value of x:
sqrt(90) - Distance from (-3,4) to P = (1/2) * Distance from P to (6,1)
sqrt(90) - sqrt((x - 6)^2 + (y - 1)^2) = (1/2) * sqrt((x + 3)^2 + (y - 4)^2)
Square both sides to eliminate the square root:
(sqrt(90) - sqrt((x - 6)^2 + (y - 1)^2))^2 = ((1/2) * sqrt((x + 3)^2 + (y - 4)^2))^2
Simplify and rearrange the equation to isolate terms:
90 - 2sqrt(90)(sqrt((x - 6)^2 + (y - 1)^2)) + (x - 6)^2 + (y - 1)^2
= (1/4)((x + 3)^2 + (y - 4)^2)
Expand and combine like terms:
90 - 2(sqrt(90))(sqrt((x - 6)^2 + (y - 1)^2)) + (x - 6)^2 + (y - 1)^2
= (1/4)((x + 3)^2 + (y - 4)^2)
Rearrange the equation:
360 - 8(sqrt(90))(sqrt((x - 6)^2 + (y - 1)^2)) + 4(x - 6)^2 + 4(y - 1)^2
= (x + 3)^2 + (y - 4)^2
Expand and simplify the equation further:
360 + 4x^2 - 48x + 144 - 16xy + 16y + 4y^2 - 8(sqrt(90))(sqrt((x - 6)^2 + (y - 1)^2))
= x^2 + 6x + 9 + y^2 - 8y + 16
Combine like terms:
3x^2 - 54x + 27 - 16xy + 20y + y^2 - 8(sqrt(90))(sqrt((x - 6)^2 + (y - 1)^2))
= 0
5. This equation represents a quadratic equation with two variables, x and y. To solve for the values of x and y, you can use numeric methods or graphing techniques, as finding an exact solution might be complex algebraically.
You may use mathematical software such as WolframAlpha, Autodesk Graphing Calculator, or programming languages like Python with the SymPy library to solve the equations numerically or graphically.
Alternatively, you may use a graphical method by plotting the given points on a graph and estimating the coordinates of the point that divides the segment in a ratio of 1:2.