Given FH with coordinates F(-7,4) and HI(-4-5), if G lies on FI such that that the ratio of FG to FH is 1:2, find the coordinates of G
Let's first find the coordinates of I. The x-coordinate of I is -4 and the y-coordinate of I is -5.
Now, let's find the x-coordinate of G. Since the ratio of FG to FH is 1:2, the difference between the x-coordinates of G and F is 1/3 of the difference between the x-coordinates of H and F. So, the x-coordinate of G is:
x-coordinate of G = x-coordinate of F + (1/3)(x-coordinate of H - x-coordinate of F)
x-coordinate of G = (-7) + (1/3)(-4 - (-7))
x-coordinate of G = -7 + (1/3)(-3)
x-coordinate of G = -7 + (-1)
x-coordinate of G = -8
Next, let's find the y-coordinate of G. Since the ratio of FG to FH is 1:2, the difference between the y-coordinates of G and F is 1/3 of the difference between the y-coordinates of H and F. So, the y-coordinate of G is:
y-coordinate of G = y-coordinate of F + (1/3)(y-coordinate of H - y-coordinate of F)
y-coordinate of G = 4 + (1/3)(-5 - 4)
y-coordinate of G = 4 + (1/3)(-9)
y-coordinate of G = 4 + (-3)
y-coordinate of G = 1
Therefore, the coordinates of G are (-8, 1).
To find the coordinates of point G, we need to determine the coordinates of point I and then find point G which lies on the line segment FI.
Given:
F(-7, 4)
H(-4, -5)
Step 1: Find the coordinates of point I
To find the coordinates of point I, we need to apply the distance formula between points F and H.
The distance formula is:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Substituting the coordinates:
d = sqrt((-4 - (-7))^2 + (-5 - 4)^2)
= sqrt((3)^2 + (-9)^2)
= sqrt(9 + 81)
= sqrt(90)
Therefore, the distance between points F and H is sqrt(90).
Since we know that ratio of FG to FH is 1:2, we can find the length of FG. Let's call FG as "d1".
d1 = (1/3) * sqrt(90)
d1 = sqrt(90) / 3
Step 2: Find the coordinates of point G
To find the coordinates of point G, we need to move a distance of d1 from point F towards point I along the line segment FI.
The x-coordinate of G can be found by subtracting d1 from the x-coordinate of F, and the y-coordinate can be found by subtracting d1 from the y-coordinate of F.
Therefore, the coordinates of point G are:
G(x, y) = (F(x) - d1, F(y) - d1)
= (-7 - sqrt(90)/3, 4 - sqrt(90)/3)
Thus, the coordinates of point G are (-7 - sqrt(90)/3, 4 - sqrt(90)/3).
To find the coordinates of point G, we need to determine the position of point I and then calculate the coordinates of G using the given information about the ratio of FG to FH.
Given points F(-7,4) and HI(-4,-5), we can find the coordinates of point I by considering the midpoint of points F and H. The midpoint formula states that the coordinates of the midpoint between two points (x₁, y₁) and (x₂, y₂) are given by:
Midpoint = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
Applying the midpoint formula to points F and H, we have:
Midpoint = ((-7 + (-4)) / 2, (4 + (-5)) / 2)
= (-11/2, -1/2)
Therefore, point I has coordinates (-11/2, -1/2).
Now, we need to find the position of G along FI, such that the ratio of FG to FH is 1:2. Since the ratio is given, we can use the section formula to find the coordinates of G.
The section formula states that if we have a line segment AB and a point C dividing AB in the ratio m:n, then the coordinates of point C are given by:
Cx = ((n * Ax) + (m * Bx)) / (m + n)
Cy = ((n * Ay) + (m * By)) / (m + n)
Applying the section formula to points F, I, and G with the ratio 1:2, we get:
Gx = ((2 * (-11/2)) + (1 * (-4))) / (1 + 2)
= (-11 + (-4)) / 3
= -15/3
= -5
Gy = ((2 * (-1/2)) + (1 * (-5))) / (1 + 2)
= (-1 + (-5)) / 3
= -6/3
= -2
Therefore, point G has coordinates (-5, -2).