The line joining (-3,10) and (4,3) is divided in what ratio by the point (-1,8)?
Ratio of x is (-3 - -1) : (-1 - 4)
= -2 : -5 or 2:5
Ratio of y is (10 - 8) : (8 - 3)
= 2:5
The ratio is 2 : 5
from (-3,10 to (-1,8)
length is sqrt(2^2 +2^2=sqrt8
from (-1,8) to (4,3)
length is sqrt (5^2 + 5^2)=sqrt50
ratio: sqrt (8/50)=2/5 sqrt(2/2)=2/5
Well, let's do some math and put on our mathematical clown noses. The first thing we can do is find the ratio that the line is divided in by the point (-1,8).
To find the ratio, we need to calculate the distances from (-3,10) to (-1,8) and from (-1,8) to (4,3). Let's call the distance from (-3,10) to (-1,8) "d1" and the distance from (-1,8) to (4,3) "d2".
Using the distance formula, d = √((x2-x1)^2 + (y2-y1)^2), we get:
d1 = √((-1 - (-3))^2 + (8 - 10)^2) = √(2^2 + (-2)^2) = √(4 + 4) = √8
d2 = √((4 - (-1))^2 + (3 - 8)^2) = √(5^2 + (-5)^2) = √(25 + 25) = √50
Now, let's find the ratio "r" by dividing d1 by d2: r = d1 / d2 = √8 / √50
But wait! We can make this ratio more fun. Let's simplify it. We notice that both √8 and √50 have the same square root at the bottom, which is √2. So, our ratio is: √8 / √50 = (√(4 * 2)) / (√(25 * 2)) = (2√2) / (5√2).
And voila! The line joining (-3,10) and (4,3) is divided in the ratio of 2√2 : 5√2 by the point (-1,8).
I hope I didn't clown around too much with the math!
To find the ratio in which the point (-1,8) divides the line segment joining (-3,10) and (4,3), we can use the section formula.
The section formula states that a point (x,y) divides a line segment joining points (x₁,y₁) and (x₂,y₂) in the ratio m:n, where m and n are positive integers, if the following equation holds:
(x,y) = ((m*x₂ + n*x₁)/(m+n), (m*y₂ + n*y₁)/(m+n))
In this case, we have:
(x,y) = (-1,8)
(x₁,y₁) = (-3,10)
(x₂,y₂) = (4,3)
Substituting these values into the section formula equation, we can solve for m:n.
(-1,8) = ((m*4 + n*(-3))/(m+n), (m*3 + n*10)/(m+n))
Simplifying further:
-1 = (4m - 3n)/(m+n)
8 = (3m + 10n)/(m+n)
We can now solve these two equations simultaneously. Let's first cross-multiply the two equations:
-1(m+n) = 4m - 3n
8(m+n) = 3m + 10n
Expanding these equations:
-m - n = 4m - 3n
8m + 8n = 3m + 10n
Rearranging the second equation:
8m - 3m = 10n - 8n - 8n
5m = 2n
Substituting this into the first equation:
-m - n = 4m - 3n
-m - n = 4(m/2) - 3(n/2)
-m - n = 2m - (3/2)n
2m + n = 0
Now, let's solve these two equations simultaneously. Rearranging the second equation:
n = -2m
Substituting this into the first equation:
2m + (-2m) = 0
0 = 0
This indicates that m and n can take any value, as long as they are positive integers, since the equation is satisfied regardless of their values.
Therefore, the point (-1,8) divides the line segment joining (-3,10) and (4,3) in any ratio, as long as m and n are positive integers.
To find the ratio in which a line is divided by a point, we can use the section formula. The section formula states that if a point (x, y) divides a line joining the points (x1, y1) and (x2, y2) in the ratio m:n, then the coordinates of the point (x, y) can be found using the following formula:
x = (n * x1 + m * x2) / (m + n)
y = (n * y1 + m * y2) / (m + n)
In this case, the two given points are (-3, 10) and (4, 3), and the dividing point is (-1, 8). Let's label the dividing point as (x, y) and the unknown ratio as m:n. We can set up the section formula as follows:
x = (n * x1 + m * x2) / (m + n)
y = (n * y1 + m * y2) / (m + n)
Plugging in the given values:
-1 = (n * -3 + m * 4) / (m + n)
8 = (n * 10 + m * 3) / (m + n)
Now, we have a system of two equations with two variables (m and n). We can solve this system of equations to find the values of m and n.
Let's multiply both sides of the first equation by (m + n) to eliminate the denominator:
-1 * (m + n) = (n * -3 + m * 4)
Expanding the equation:
-m - n = -3n + 4m
Rearranging the terms:
4m + n = 3n - m
Now, let's multiply both sides of the second equation by (m + n):
8 * (m + n) = (n * 10 + m * 3)
Expanding the equation:
8m + 8n = 10n + 3m
Rearranging the terms:
8m - 3m = 10n - 8n
5m = 2n
Now, we have two equations:
4m + n = 3n - m
5m = 2n
We can solve these equations simultaneously using any method, such as substitution or elimination, to find the values of m and n.