find the point which divides the segment from (-5,-4) to (6,-2) in the ratio 2:3
make a sketch and let our point needed be (x,y)
for the x:
(x-(-5))/(6-x) = 2/3
3x + 15 = 12-2x
5x = -3
x = -3/5
for the y:
(y+4)/(-2-y) = 2/3
3y + 12 = -4-2y
5y = -16
y = -16/5
your point is (-3/5 , -16/5)
or, using vectors
(x,y) = (3/5)(-5,-4) + (2/5)(6,-2)
= (-3 , -12/5) + (12/5, -4/5)
= (-3/5 , -16/5)
Well, let's divide this question into two parts: finding the x-coordinate and the y-coordinate of the point.
First, let's find the x-coordinate. The ratio 2:3 means that the point is two parts of the way from (-5,-4) to (6,-2).
So, the total distance in the x-direction is 6 - (-5) = 11. Two parts of this distance would be (2/5) * 11 = 4.4.
Starting from -5, adding 4.4 would give us -5 + 4.4 = -0.6. Therefore, the x-coordinate is approximately -0.6.
Now, let's move on to finding the y-coordinate. Again, the ratio 2:3 means that the point is two parts of the way from -4 to -2.
The total difference in the y-direction is -2 - (-4) = 2. Two parts of this distance would be (2/5) * 2 = 0.8.
Starting from -4, adding 0.8 gives us -4 + 0.8 = -3.2. Therefore, the y-coordinate is approximately -3.2.
Putting it all together, the point that divides the segment from (-5,-4) to (6,-2) in a 2:3 ratio is approximately (-0.6, -3.2).
Now, that wasn't too dividing, was it?
To find the point that divides the segment from (-5,-4) to (6,-2) in the ratio 2:3, follow these steps:
Step 1: Calculate the difference between the x-coordinates and y-coordinates of the two given points.
Δx = (6 - (-5)) = 11
Δy = (-2 - (-4)) = 2
Step 2: Find the total length of the segment.
d = sqrt(Δx^2 + Δy^2) = sqrt(11^2 + 2^2) = sqrt(121 + 4) = sqrt(125) = 11.18
Step 3: Determine the coordinates of the point using the ratio.
Let the coordinates be (x, y).
The point divides the segment into two parts: one with length (2/5) * d and the other with length (3/5) * d.
The coordinates of the point can be calculated as follows:
x = (-5 + (3/5) * 11) = (-5 + 6.6) = 1.6
y = (-4 + (3/5) * 2) = (-4 + 1.2) = -2.8
Therefore, the point that divides the segment from (-5,-4) to (6,-2) in the ratio 2:3 is (1.6, -2.8).
To find the point that divides the segment from (-5, -4) to (6, -2) in the ratio 2:3, follow these steps:
Step 1: Calculate the distance between the two points.
The distance formula is given by:
d = sqrt( (x2 - x1)^2 + (y2 - y1)^2 )
Using the formula, we get:
d = sqrt( (6 - (-5))^2 + (-2 - (-4))^2 )
= sqrt(11^2 + 2^2)
= sqrt(121 + 4)
= sqrt(125)
= 5 * sqrt(5)
Therefore, the distance between (-5,-4) and (6,-2) is 5 * sqrt(5).
Step 2: Use the ratio 2:3 to find the coordinates of the desired point.
Let's assume the desired point is P(x,y) and the ratio of 2:3 divides the segment in terms of distance.
The x-coordinate of P can be found using the following formula:
x = ( (x2 * m) + (x1 * n) ) / (m + n)
where x1 and x2 are the x-coordinates of the given points, and m and n are the ratio terms (2 and 3, respectively).
Substituting the values, we get:
x = ( (6 * 2) + (-5 * 3) ) / (2 + 3)
= (12 - 15) / 5
= -3 / 5
The y-coordinate of P can be found similarly:
y = ( (y2 * m) + (y1 * n) ) / (m + n)
Substituting the values, we get:
y = ( (-2 * 2) + (-4 * 3) ) / (2 + 3)
= (-4 - 12) / 5
= -16 / 5
Therefore, the point that divides the segment from (-5,-4) to (6,-2) in the ratio 2:3 is (-3/5, -16/5).