Section Formulae worksheets are available here. The students of grade 10 can find various problems on the Worksheet on Section Formula. There are a lot of uses for solving the questions related to section formula, one is you can know how to solve the given question in time with a simple method and the other practice for the exams and improve the math skills that help for you in further education. Tap the solution link to get the step by step explanation for all the questions on section formula to test yourself.
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Section Formula Questions with Solutions
Example 1.
Using section formula, show that the points A(7,5) B(9,3), and C(3,1) are collinear.
Solution:
If three points are collinear, then one of the points divides the line segment joining the other points in the ratio r : 1.
If P is between A and B and AP/PB = r
Distance AB = √(x2 – x1)² + (y2 – y1)²
= √(9 – 7)² + (3 + 5)²
= √(2)² + (8)²
= √4 + 64
= √68 = 2√17
Distance BC = √(3 -9)² + (1 – 3)²
= √(-6)² + (-2)²
= √36 + 4
= √40
= 2√10
To find r
r = AB/BC = 2√17/2√10
= 4.1 / 3.1
The line divides in the ratio 4.1 : 3.1 A line divides internally in the ratio m:n.
The point P = (mx2 + nx1/m + n, my2 + ny2/m + n)
The point B = (4.1×3 + 3.1 × 7/4.1 + 3.1, 4.1×1 + 3.1×5/4.1 + 3.1)
B = (34/7.2, 19.6/7.2)
B = (4.7, 2.7)
Example 2.
The 4 vertices of a parallelogram are A(-2, 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(-2, 3), B(3, -1), C(p, q) and D(-1, 9)
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)
-2 + p/2 = -1 + 3/2
-2 + P = 2
P = 2 + 2
P = 4
Similarly
3 + q/2 = 9 – 1/2
3 + q = 8
q = 3 – 8
q = 5
Example 3.
Find the coordinates of the point which divides the line segment joining the points (2,4) and (3,5) internally in the ratio 3:2.
Solution:
Let P(x, y) be the point which divides the line segment joining A(2,4) and B(3,5) internally in the ratio 3 : 2.
Here,
(x1, y1) = (2,4)
(x2, y2) = (3,5)
m : n = 3 : 2
Using the section formula,
P(x, y) = (mx1+nx2/m + n , my1+my2/m + n)
(3×3 + 2×2/3 + 2, 3×5 + 2×4/3 + 2)
(9 + 4/5, 15 + 8/5)
(13/5, 23/5)
x = 13/5, y = 23/5
Example 4.
Z (4, 5) and X(7, – 1) are two given points and the point Y divides the line-segment ZX externally in the ratio 4:3. Find the coordinates of Y.
Solution:
Given that, Z(4,5)=(x1,y1), X(7,-1)=(x2,y2)
Point Y divides the segment ZX in the ratio 4:3, hence m=4, n=3
Since it is mentioned in the question that the point Y divides the segment externally we use the section formula for external division,
Section formula Y= [(mx2-nx1)/(m-n),(my2-ny1)/(m-n)]
Substituting the known values,
=[(4(7)-3(4))/(4-3),(4(-1)-3(5)/(4-3)]
=((28-12)/1,(-4-15)/1)
=(16,-19)
The coordinates for the point Y are (16,-19)
Example 5.
Find the coordinates of the point M which divides a line segment PQ in the ratio 2:3. PQ line segment joins the points P(1,2) and Q(3,4) does the point M lie on the line 5y – x = 20.
Solution:
Given that
P(1,2) and Q(3,4)
Let the x coordinates of M be x and y coordinates be y : M(x,y).
m:n = 2:3
We know that section formula
M(x,y) = M(mx2 + nx2/m + n, my2 + ny1/m + n)
M(x,y) = M(2(3) + 3(1)/2+3, 2(4) + 3(2)/2+3)
= M(6+3/ 5, 8+6/5)
= M(9/5, 14/5)
M(x, y) = (9/5,14/5)
x = 9/5, y = 14/5
Hence the points M is (9/5,14/5)
Putting the values of x and y in 5y – x = 20
5(14/5) – (9/5) – 20 = 0
Hence points lie on the line 5y – x = 20
Example 6.
Given a line segment AB joining the points A(-4,3) and B(8,-3) find the relation in which AB is divided by the y axis. And find the coordinates of the point of intersection.
Solution:
Let the required ratio be m1 : m2
Consider A(-4,3) = (x1,y1)
B(8,-3) = (x2, y2)
Let P(x,y) be the point and intersection of the line segment the y axis.
By section formula we have
x = m1x2 + m2x1/m1 + m2,
y = m1y2 + m2y1/m1 + m2
x = 8m1 – 4m2/m1 + m2
y = -3m1 + 3m2/m1 + m2
The equation of the y axis is x = 0
x = 8m1 – 4m2/m1 + m2 = 0
8m1 – 4m2 = 0
8m1 = 4m2
m1/m2 = 4/8
m1/m2 = 1/2
Know we consider
m1 = k and m2 = 2k
x = 8m1 – 4m2/m1 + m2
y = -3m1 + 3m2/m1 + m2
x = 8k – 4× 2k/k + 2k
y = -3k + 3 × 2k/k + 2k
x = 8k – 8k/3k, y = -3k + 6k/3k
x = 0/3k
y = 3k/3k
(x,y) =( 0,1)
Thus the point intersection is P(0,1)
Example 7.
Find the coordinates of the points of trisection of the line segment joining the point (6,4) and the origin.
Solution:
Let P and Q be the points at trisection of the line segment joining A(6,4) and B(0,0). P provides AB in the ratio 1:2
Therefore the coordinates of the point P are.
(1×0 + 2×6/1 + 2, 1×0 + 2×4/1 + 2)
(0+12/3,1+8/3
(4,3)
Q divides AB in the ratio 2 : 1 therefore the coordinates of point Q are
(2×0 + 1×6/2+1, 2×0 + 1×4/2+1)
(6/3, 4/3)
(2,4/3)
Thus the required points are (4,3) and (2,4/3)
Example 8.
A line segment joining A(-1,5/3) and B(a,5) is divided in the ratio 1:3 at P The point where the line segment AB intersects the y axis.
i) calculate the value of a.
ii) calculate the coordinates of P.
Solution:
The line segment AB intersects the y axis at a point in P, let the coordinates of point p be (0,y).
P divides AB in the ratio 1:3.
(0,y) = (1×a + 3(-1)/1+3, 1×5 + 3 + 5/3/1+3)
(0,y) = (a-3/ 4, 1//4)
0 = a-3/ 4 and y = 10/4
a = 3, and y = 5/2.
Thus the value of a is 3 and the coordinates of point P are (0,5/2)
Example 9.
In what ratio is the line joining A(0,4) and B(2,-1) divided by the x axis write the coordinates of the point where AB intersects the x axis.
Solution:
Let the line segment AB intersects the x axis by point P(x,0) in the ratio k:1.
(x,0) = (k×2+1×0/k+1, k(-1)+1×4/k+1)
= 2k/k+1, -k+4/k+1
0 = -k+4/k+1
k = 4.
Thus the required ratio in which P divides AB is 4:1.
x = 2k/k+1
x = 2(4)/4+1 = 8/5.
Thus the coordinates of point P are (3,0)
Example 10.
At the midpoint of the segments AB and C(3,4) write the coordinates of A and B.
Solution:
Consider A lies on the x axis and let the coordinates of point A be (x,0).
Consider point B lies on y axis, let the coordinates of the point B be (0, y)
Given mid point of AB is C(3,4)
(3,4) = (x+0/2, 0+y/2)
(3,4) = (x/2, y/2)
3 = x/2 and 4 = y/2
x = 6, and y = 8.
Thus the coordinates of point A are (6,0) and the coordinates of point B are (0,8)