Are you browsing various sites to learn about section formulae and distance formulae? Then have a look at this article to know briefly about the section formula and the formulas used with examples. Usually, the section formulae help to find the coordinates of the point to divide line joining points in a ratio. Scroll down this page to find the derivation of the section formula.

## Section Formula

In geometry, the section formula is used to find the ratio in which the line segment is divided by point externally or internally. We use a section formula to find the incenter, centroid, and excenter of the triangle. We can find the midpoint of the line segment with the help of the section formula.

### Section Formula Derivation

Let P(x1, y1) and Q(x2, y2) be two points in the above XY-plane.
Let M(x, y) divide the line segment PQ equally with the ratio m:n.
PA, MN and QR are drawn parallel to x-axis.
∠MPS = ∠QMB (corresponding angles)
∠MSP = ∠QBM = 90°
By following AA similarity
ΔPMS ~ ΔMQB
PM/MQ = PS/MB = MS/QB = m/n
PS = AN = ON – OA = x – x1
MB = NR = OR – ON = x2 – x
MS = MN – SN = y – y1
QB = RQ – RB = y2 – y
m/n = x – x1/x2 – x = y – y1/y2 – y
x = mx2 + nx1/m + n
m/n = y – y1/y2 – y
y = my2 + ny1/m + n
Section Formula (Internally):
m:n = (mx2 + nx1/m + n, my2 + ny1/m + n)
Section Formula (Externally):
m:n = (mx2 – nx1/m – n, my2 – ny1/m – n)

### Section Formula Examples

Example 1.
Find the coordinates of the point which divides the line segment joining the points (2,1) and (3,1) internally in the ratio 3:2.
Solution:
Let P(x, y) be the point which divides the line segment joining A(2,1) and B(3,1) internally in the ratio 2 : 1
Here,
(x1, y1) = (2,1)
(x2, y2) = (3,1)
m : n = 2 : 1
Using the section formula,
P(x, y) = (mx1+nx2/m + n , my1+my2/m + n)
(2×1 + 1×3/2 + 1, 2×1 + 1×1/2 + 1)
(2 + 3/3, 2 + 1/3)
(5/3, 3/3)
(5/3,1)
x = 5/3, y = 1

Example 2.
Find the coordinates of the points of trisection of the line segment joining the point (3,2) and the origin.
Solution:
Let P and Q be the points at trisection of the line segment joining A(3,2) and B(0,0). P provides AB in the ratio of 1:2
Therefore the coordinates of the point P are.
(1×0 + 2×3/1 + 2, 1×0 + 2×2/1 + 2)
(0+6/3,4/3
(2,4/3)
Q divides AB in the ratio 2: 1 therefore the coordinates of point Q are
(2×0 + 1×3/2+1, 2×0 + 1×2/2+1)
(3/3, 2/3)
(1,2/3)
Thus the required points are (3,2) and (1,4/3)

Example 3.
Given a triangle ABC in which A = (4,-4) B(0,5) and C(5,10). A point P lies on BC such that BP : PC = 3:2. Find the length of line segment AP.
Solution:
Given that
BP:PC = 3:2
Using the section formula the coordinates of a point P are
(3×5+2×0/3+2, 3×10+2×5/3+2)
(15/2,40/5)
(3,8)

Example 4.
The 4 vertices of a parallelogram are A(-4, 3), B(3, -1), C(p, q), and D(-1, 9). Find the value of p and q.
Solution:
Given vertices of a parallelogram are:
A(-4, 3), B(3, -1), C(p, q) and D(-1, 3)
We know that diagonals of a parallelogram bisect each other.
Let O be the point at which diagonals intersect.
Coordinates of mid-points of both AC and BD will be the same.
Therefore,
Using midpoint section formula,
(x1+x2/2 , y1+y2/2)
-4 + p/2 = -1 + 3/2
-4 + P = 2
P = 4 + 2
P = 6
Similarly
3 + q/2 = 3 – 1/ 2
3 + q = 2
q = 2-3
q = -1

Example 5.
Find the coordinates of the point which divides the line segment joining the points (4,2) and (5,1) internally in the ratio 3:2.
Solution:
Let P(x, y) be the point which divides the line segment joining A(4,2) and B(5,1) internally in the ratio 2 : 1
Here,
(x1, y1) = (4,1)
(x2, y2) = (5,1)
m : n = 2 : 1
Using the section formula,
P(x, y) = (mx1+nx2/m + n , my1+my2/m + n)
(2×4 + 1×5/2 + 1, 2×2 + 1×1/2 + 1)
(8 + 5/3, 4 + 1/3)
(13/3, 5/3)
(14/3,5/3)
x = 14/3, y = 5/3

### FAQs on Section Formula

1. What are the Applications of Section Formula?

Section formula is used in various places in mathematics. In mathematics, we can use the section formula to find the centroid, incenters, and excenters of a triangle, etc. The section formula is also widely used to find the midpoint of a line segment.

2. What is the Section Formula for Internal Division?

If we have a line segment AB that is divided by a point P(x, y) internally in a ratio such that AP: PB = m: n,
Then the section formula for internal division is:
P(x, y) = (mx2+nx1/m+n,my2+ny1/m+n)
where,
x and y are the coordinates of point P
(x1, y1) are the coordinates of point A
(x2, y2) are the coordinates of the point B
m and n are the ratio values in which P divides the line internally

3. How is the section formula derived?

Section formula can be derived by constructing two right triangles and by using AA similarity. To find the ratio of the length of the sides of the triangle solve for x, and y, we can find the coordinates of the point that is dividing the line segment.