Section Formula Calculator

About Section Formula Calculator

The section formula divides a segment in ratio m:n. Internal division: point lies between endpoints. External: outside.

How to use this calculator

1. Enter the coordinates of the two endpoints of the segment.
2. Enter the ratio in the form m:n, using the parts on each side of the dividing point.
3. Calculate the weighted average for each coordinate to find the point.
4. Check that the point lies on the segment and matches the given ratio.

The formula explained

• \(x\) = the x-coordinate of the point that divides the segment
• \(y\) = the y-coordinate of the point that divides the segment
• \(x_1\) = the x-coordinate of the first endpoint
• \(y_1\) = the y-coordinate of the first endpoint
• \(x_2\) = the x-coordinate of the second endpoint
• \(y_2\) = the y-coordinate of the second endpoint
• \(m\) = the ratio part from the first endpoint to the dividing point
• \(n\) = the ratio part from the dividing point to the second endpoint

Step by step method

1. Identify the two endpoints and label them \(A(x_1, y_1)\) and \(B(x_2, y_2)\).
2. Write the ratio as \(m:n\), where \(m\) corresponds to the part from the first endpoint and \(n\) to the part from the second endpoint.
3. Substitute the values into \(x = \frac{m x_2 + n x_1}{m + n}\) and \(y = \frac{m y_2 + n y_1}{m + n}\).
4. Simplify the fractions to get the coordinates of the dividing point.

Worked example

Suppose a point divides the segment from A(2, 4) to B(8, 10) in the ratio 1:2.

1. Use \(x = \frac{m x_2 + n x_1}{m + n}\) and \(y = \frac{m y_2 + n y_1}{m + n}\) with \(m = 1\), \(n = 2\), \(x_1 = 2\), \(y_1 = 4\), \(x_2 = 8\), and \(y_2 = 10\).
2. Compute the x-coordinate: \(x = \frac{1 \cdot 8 + 2 \cdot 2}{1 + 2} = \frac{8 + 4}{3} = 4\).
3. Compute the y-coordinate: \(y = \frac{1 \cdot 10 + 2 \cdot 4}{1 + 2} = \frac{10 + 8}{3} = 6\).

Tips and common mistakes

• Make sure the ratio parts match the correct endpoints. Swapping them gives the wrong point.
• Use the same ratio for both coordinates. The x and y values must come from the same weighted average.
• If the ratio is 1:1, the point is the midpoint.
• Double-check the order of the endpoints before substituting values.

Frequently asked questions