Find the coordinates of the point which divides the line segment joining (2,4) and (6,8) in the ratio 1:3 internally and externally
so it is 1/4 the way, right?
x'=2+(6-2)/4=3
y'=4+(8-4)/4=5
LM has the endpoints L at –5 and M at 9. To find the point x so that x divides the directed line segment LM in a 2:3 ratio, use the formula x = (x2 – x1) + x1.
x = (9 – (–5)) + (–5)
To find the coordinates of the point that divides the line segment joining two points in a specific ratio, we can use the section formula.
1. To find the point internally, let's first find the coordinates of the point that divides the line segment in the ratio 1:3 internally.
The section formula for internal division is:
\[ P = \left(\frac{(x_1 \cdot m + x_2 \cdot n)}{(m + n)}, \frac{(y_1 \cdot m + y_2 \cdot n)}{(m + n)}\right) \]
Where (x1, y1) and (x2, y2) are the coordinates of the two given points, and m:n is the ratio in which the line segment is divided.
Given points: (2, 4) and (6, 8)
Ratio: 1:3 (internal division)
Let's substitute the values into the formula:
\[ P = \left(\frac{(2 \cdot (\frac{1}{4}) + 6 \cdot (\frac{3}{4}))}{(\frac{1}{4} + \frac{3}{4})}, \frac{(4 \cdot (\frac{1}{4}) + 8 \cdot (\frac{3}{4}))}{(\frac{1}{4} + \frac{3}{4})}\right) \]
Simplifying the equation:
\[ P = \left(\frac{(2 + 18)}{4}, \frac{(1 + 24)}{4}\right) \]
\[ P = \left(\frac{20}{4}, \frac{25}{4}\right) \]
\[ P = \left(5, \frac{25}{4}\right) \]
Therefore, the coordinates of the point that divides the line segment joining (2,4) and (6,8) in the ratio 1:3 internally are (5, 25/4).
2. To find the point externally, let's find the coordinates of the point that divides the line segment in the ratio 1:3 externally.
The section formula for external division is:
\[ P = \left(\frac{(x_1 \cdot m - x_2 \cdot n)}{(m - n)}, \frac{(y_1 \cdot m - y_2 \cdot n)}{(m - n)}\right) \]
Given points: (2, 4) and (6, 8)
Ratio: 1:3 (external division)
Let's substitute the values into the formula:
\[ P = \left(\frac{(2 \cdot (\frac{1}{4}) - 6 \cdot (\frac{3}{4}))}{(\frac{1}{4} - \frac{3}{4})}, \frac{(4 \cdot (\frac{1}{4}) - 8 \cdot (\frac{3}{4}))}{(\frac{1}{4} - \frac{3}{4})}\right) \]
Simplifying the equation:
\[ P = \left(\frac{(2 - 18)}{4}, \frac{(1 - 24)}{4}\right) \]
\[ P = \left(\frac{-16}{-2}, \frac{-23}{-2}\right) \]
\[ P = \left(8, \frac{23}{2}\right) \]
Therefore, the coordinates of the point that divides the line segment joining (2,4) and (6,8) in the ratio 1:3 externally are (8, 23/2).
To find the coordinates of the point that divides the line segment internally and externally in a given ratio, we can use the section formula.
1. For internal division in the ratio 1:3:
Let the coordinates of the point be (x, y).
Using the section formula:
x = (3 * 2 + 1 * 6) / (3 + 1) = (6 + 6) / 4 = 12 / 4 = 3
y = (3 * 4 + 1 * 8) / (3 + 1) = (12 + 8) / 4 = 20 / 4 = 5
Therefore, the coordinates of the point internally dividing the line segment in a 1:3 ratio are (3, 5).
2. For external division in the ratio 1:3:
Let the coordinates of the point be (x, y).
Using the section formula:
x = (3 * 2 - 1 * 6) / (3 - 1) = (6 - 6) / 2 = 0 / 2 = 0
y = (3 * 4 - 1 * 8) / (3 - 1) = (12 - 8) / 2 = 4 / 2 = 2
Therefore, the coordinates of the point externally dividing the line segment in a 1:3 ratio are (0, 2).