Help!!! Line segment EF in the xy-coordinate plane has endpoints with coordinates (-4,9) and (6,4). What are two possible locations for point G so that G divides EF into two parts with lengths in a ratio of 1:4?
Just like all the related questions below,
The x,y distances from E to F are (10,-5)
If G is 1/5 of the way from E to F, add 1/5 of the difference to E.
G = (-4,9)+(2,-1) = (-2,8)
If G is at the other end, subtract those values from F.
To find the two possible locations for point G, we need to divide the line segment EF into two parts with lengths in a ratio of 1:4.
First, let's find the length of the line segment EF. We can use the distance formula, which states that the distance between two points (x₁, y₁) and (x₂, y₂) is given by:
Distance = √((x₂ - x₁)² + (y₂ - y₁)²)
Using the coordinates (-4, 9) and (6, 4), we can calculate the distance EF:
Distance = √((6 - (-4))² + (4 - 9)²)
= √(10² + (-5)²)
= √(100 + 25)
= √125
= 5√5
Now, since we want the ratio of the lengths of EG and FG to be 1:4, let's assign lengths to these two parts. Let EG have a length of x, and FG have a length of 4x.
To divide EF into these two parts in the given ratio, we can use the concept of section formula.
The coordinates of point G can be found by using the section formula, which states that for a line segment with endpoints (x₁, y₁) and (x₂, y₂), dividing it into parts in the ratio of m:n (where m + n = total parts), the coordinates of the dividing point are given by:
(x, y) = ( (x₂ * m + x₁ * n) / (m + n), (y₂ * m + y₁ * n) / (m + n) )
In our case, dividing EF into parts in a ratio of 1:4, we have:
(x, y) = ( (6 * 1 + (-4) * 4) / (1 + 4), (4 * 1 + 9 * 4) / (1 + 4) )
Simplifying this expression, we get:
(x, y) = ( (-10) / 5 , (40 + 36) / 5 )
= (-2, 16/5)
= (-2, 3.2)
So, one possible location for point G is (-2, 3.2).
Similarly, we can find the coordinates for the other possible location for point G by using the ratio 4:1 (since we want EG to be 4 times longer than FG):
(x, y) = ( (6 * 4 + (-4) * 1) / (4 + 1), (4 * 4 + 9 * 1) / (4 + 1) )
Simplifying this expression, we get:
(x, y) = ( (24 - 4) / 5 , (16 + 9) / 5 )
= (20/5, 25/5)
= (4, 5)
So, the other possible location for point G is (4, 5).
Therefore, the two possible locations for point G are (-2, 3.2) and (4, 5).