A directed line segment begins at F(-10, -4), ends at H(8, 4), and is divided in the ratio 3 to 1 by G.
so G is 3/4 of the way from F to H
Oh, the tale of a directed line segment and its adventurous journey from F to H, with a mischievous character named G dividing it in a 3 to 1 ratio. Quite the mathematical drama!
Now, let's see where this merry band of coordinates takes us. We have F(-10, -4) on one end, H(8, 4) on the other, and G somewhere in the middle, playing referee.
To find the coordinates of G, we'll use the concept of dividing the line segment in a given ratio. Since G is dividing the line segment in a 3 to 1 ratio, we can think of it as three parts heading towards F and one part heading towards H.
First, let's find the distance between F and H in both the x-axis and the y-axis:
The horizontal distance (change in x) is 8 - (-10) = 18.
The vertical distance (change in y) is 4 - (-4) = 8.
Since we want a 3 to 1 ratio, we know that 3 parts will go towards F (18 * 3/4), and 1 part will go towards H (18 * 1/4).
For the x-coordinate:
3 parts towards F = 18 * 3/4 = 13.5.
So, G's x-coordinate will be F's x-coordinate (starting point) minus 13.5:
-10 - 13.5 = -23.5.
For the y-coordinate:
3 parts towards F = 8 * 3/4 = 6.
So, G's y-coordinate will be F's y-coordinate (starting point) minus 6:
-4 - 6 = -10.
So, with all this mathematical tomfoolery, we find that G's coordinates are approximately (-23.5, -10). Quite the oddball coordinates, isn't it? But hey, that's what you get when you let a clown like me do the math!
To find the coordinates of point G, we can use the concept of the section formula. The section formula states that if a line segment with endpoints A(x1, y1) and B(x2, y2) is divided by a point G in the ratio m:n, then the coordinates of point G can be obtained using the following formula:
G(x, y) = ( (n * x1 + m * x2) / (m + n) , (n * y1 + m * y2) / (m + n) )
In this case, the line segment starts at F(-10, -4) and ends at H(8, 4), and it is divided in a 3:1 ratio by point G.
Plugging in the values, we get:
G(x, y) = ( (1 * -10 + 3 * 8) / (3 + 1) , (1 * -4 + 3 * 4) / (3 + 1) )
Simplifying, we have:
G(x, y) = ( ( -10 + 24 ) / 4 , ( -4 + 12 ) / 4 )
G(x, y) = ( 14 / 4 , 8 / 4 )
G(x, y) = ( 7 , 2 )
Therefore, point G has coordinates G(7, 2).
To find the coordinates of point G, we can use the concept of section formula in coordinate geometry. The section formula states that if a line segment with endpoints (x1, y1) and (x2, y2) is divided by a point (x, y) internally in the ratio of m:n, then the coordinates of the dividing point (x, y) can be found using the following formula:
x = ((m * x2) + (n * x1)) / (m + n)
y = ((m * y2) + (n * y1)) / (m + n)
In this case, we are given the endpoints F(-10, -4) and H(8, 4), and the ratio of 3 to 1. So, m = 3 and n = 1.
Using the section formula, we can find the coordinates of point G:
x = ((3 * 8) + (1 * -10)) / (3 + 1)
= (24 - 10) / 4
= 14 / 4
= 7/2
y = ((3 * 4) + (1 * -4)) / (3 + 1)
= (12 - 4) / 4
= 8 / 4
= 2
Therefore, the coordinates of point G are (7/2, 2).