Point Sections Or Ratio Formulae: 41 Critical Solutions

Basic Examples on the Formulae “Point sections or Ratio”

Case-I

Problems 21: Find the coordinates of the point P(x, y) which internally divides the line segment joining the two points (1,1) and (4,1) in the ratio 1:2.

Solution: We already know,

If a point P(x, y) divides the line segment AB internally in the ratio m:n,where coordinates of A and B are (x₁,y₁) and (x₂,y₂) respectively. Then Coordinates of P are

and

(See formulae chart)

Using this formula we can say , (x₁,y₁) ≌(1,1) i.e. x₁=1, y₁=1 ;

(x₂,y₂)≌(4,1) i.e. x₂=4, y₂=1

and

m:n ≌ 1:2 i.e m=1,n=2

Therefore,

x =

( putting values of m & n in

Or, x =1*4+2*1/3 ( putting values of x₁ & x₂ too )

Or, x = 4+2/3

Or, x = 6*3

Or, x = 2

Similarly we get,

y =

( putting values of m & n in y =

Or, y =(1*1+2*1)/3 ( putting values of y₁ & y₂ too )

Or, y = 1*1+2/3

Or, y = 3/3

Or, y = 1

Therefore, x=2 and y=1 are the coordinates of the point P i.e. (2,1). (Ans)

More answered problems are given below for further practice using the procedure described in above problem 21:-

Problem 22: Find the coordinates of the point which internally divides the line segment joining the two points (0,5) and (0,0) in the ratio 2:3.

Ans. (0,2)

Problem 23: Find the point which internally divides the line segment joining the points (1,1) and (4,1) in the ratio 2:1.

Ans. (3,1)

Problem 24: Find the point which lies on the line segment joining the two points (3,5,) and (3,-5,) dividing it in the ratio 1:1

Ans. (3,0)

Problem 25: Find the coordinates of the point which internally divides the line segment joining the two points (-4,1) and (4,1) in the ratio 3:5

Ans. (-1,1)

Problem 26: Find the point which internally divides the line segment joining the two points (-10,2) and (10,2) in the ratio 1.5 : 2.5.

_____________________________

Case-II

Problems 27: Find the coordinates of the point Q(x,y) which externally divides the line segment joining the two points (2,1) and (6,1) in the ratio 3:1.

Solution: We already know,

If a point Q(x,y) divides the line segment AB externally in the ratio m:n,where coordinates of A and B are (x₁,y₁) and (x₂,y₂) respectively,then the coordinates of the point P are

and

(See formulae chart)

Using this formula we can say , (x₁,y₁) ≌(2,1) i.e. x₁=2, y₁=1 ;

(x₂,y₂)≌(6,1) i.e. x₂=6, y₂=1 and

m:n ≌ 3:1 i.e. m=3,n=1

Point sections — Graphical Representation

Therefore,

x =

( putting values of m & n in x =

Or, x =(3*6)-(1*2)/2 ( putting values of x₁ & x₂ too )

Or, x = 18-2/2

Or, x =16/2

Or, x = 8

Similarly we get,

y =

( putting values of m & n in y =

Or, y =

( putting values of y₁ & y₂ too )

Or, y = 3-1/2

Or, y = 2/2

Or, y = 1

Therefore, x=8 and y=1 are the coordinates of the point Q i.e. (8,1). (Ans)

More answered problems are given below for further practice using the procedure described in above problem 27:-

Problem 28: Find the point which externally divides the line segment joining the two points (2,2) and (4,2) in the ratio 3 : 1.

Ans. (5,2)

Problem 29: Find the point which externally divides the line segment joining the two points (0,2) and (0,5) in the ratio 5:2.

Ans. (0,7)

Problem 30: Find the point which lies on the extended part of the line segment joining the two points (-3,-2) and (3,-2) in the ratio 2 : 1.

Ans. (9,-2)

________________________________

Case-III

Problems 31: Find the coordinates of the midpoint of the line segment joining the two points (-1,2) and (1,2).

Solution: We already know,

If a point R(x,y) be the midpoint of the line segment joining A(x₁,y₁) and B(x₂,y₂) .Then coordinates of R are

and

(See formulae chart)

Case-III is the form of case-I while m=1 and n=1

Using this formula we can say , (x₁,y₁) ≌(-1,2) i.e. x₁=-1, y₁=2 and

(x₂,y₂)≌(1,2) i.e. x₂=1, y₂=2

Screenshot 11 — Graphical Representation

Therefore,

x =

( putting values of x₁ & x₂ in x =

Or, x = 0/2

Or, x = 0

Similarly we get,

y =2+2/2 ( putting values of y₁ & y₂ in y =

Or, y = 4/2

Or, y = 2

Therefore, x=0 and y=2 are the coordinates of the midpoint R i.e. (0,2). (Ans)

More answered problems are given below for further practice using the procedure described in above problem 31:-

Problem 32: Find the coordinates of the midpoint of the line joining the two points (-1,-3) and (1,-4).

Ans. (0,3.5)

Problem 33: Find the coordinates of the midpoint which divides the line segment joining the two points (-5,-7) and (5,7).

Ans. (0,0)

Problem 34: Find the coordinates of the midpoint which divides the line segment joining the two points (10,-5) and (-7,2).

Ans. (1.5, -1.5)

Problem 35: Find the coordinates of the midpoint which divides the line segment joining the two points (3,√2) and (1,3√2).

Ans. (2,2√2)

Problem 36: Find the coordinates of the midpoint which divides the line segment joining the two points (2+3i,5) and (2-3i,-5).

Ans. (2,0)

Note: How to check if a point divides a line (length=d units) internally or externally by the ratio m:n

If ( m×d)/(m+n) + ( n×d)/(m+n) = d , then internally dividing and

If ( m×d)/(m+n) – ( n×d)/(m+n) = d , then externally dividing

____________________________________________________________________________

Basic Examples on the Formulae “Area of a Triangle”

Case-I

Problems 37: What is the area of the triangle with two vertices A(1,2) and B(5,3) and height with respect to AB be 3 units in the coordinate plane ?

Solution: We already know,

If “h” be the height and “b” be the base of Triangle, then Area of the Triangle is = ½ × b × h

(See formulae chart)

image?w=366&h=269&rev=57&ac=1&parent=1Ug0lE5AOAhO4i0HE5fVqVUKTEbR0on8yfNNyWgAF Po — Graphical Representation

Using this formula we can say ,

h = 3 units and b = √

$(x2-x1)2+(y2-y1)2 $

i.e √

$(5-1)2+(3-2)2 $

Or, b = √

$(4)2+(1)2 $

Or, b = √

$(16+1 $

Or, b = √ 17 units

Therefore, the required area of the triangle is = ½ × b × h i.e

= ½ × (√ 17 ) × 3 units

= 3⁄2 × (√ 17 ) units (Ans.)

______________________________________________________________________________________

Case-II

Problems 38:What is the area of the triangle with vertices A(1,2), B(5,3) and C(3,5) in the coordinate plane ?

Solution: We already know,

If A(x₁,y₁) , B(x₂,y₂) and C(x₃,y₃) be the vertices of a Triangle,

Area of the Triangle is =|½

$x1 (y2- y3) + x2 (y3- y2) + x3 (y2- y1)$

(See formulae chart)

Using this formula we have ,

(x₁,y₁) ≌(1,2) i.e. x₁=1, y₁=2 ;

(x₂,y₂) ≌(5,3) i.e. x₂=5, y₂=3 and

(x₃,y₃) ≌(3,5) i.e. x₃=3, y₃=5

image?w=364&h=194&rev=207&ac=1&parent=1Ug0lE5AOAhO4i0HE5fVqVUKTEbR0on8yfNNyWgAF Po — Graphical Representation

Therefore, the area of the triangle is = |½

$x1 (y2- y3) + x2 (y3- y1) + x3 (y1-y2)$

| i.e

= |½

$1 (3-5) + 5 (5-3) + 3 (3-2)$

| sq.units

= |½

$1x (-2) + (5×2) + (3×1)$

| sq.units

= |½

$-2 + 10 + 3$

| sq.units

= |½ x 11| sq.units

= 11⁄2 sq.units

= 5.5 sq.units (Ans.)

More answered problems are given below for further practice using the procedure described in above problems :-

Problem 39: Find the area of the triangle whose vertices are (1,1), (-1,2) and (3,2).

Ans. 2 sq.units

Problem 40: Find the area of the triangle whose vertices are (3,0), (0,6) and (6,9).

Ans. 22.5 sq.units

Problem 41: Find the area of the triangle whose vertices are (-1,-2), (0,4) and (1,-3).

Ans. 6.5 sq.units

Problem 42: Find the area of the triangle whose vertices are (-5,0,), (0,5) and (0,-5). Ans. 25 sq.units

_______________________________________________________________________________________

For more post on Mathematics, please follow our Mathematics page.

NASRINA PARVIN

Hi….I am Nasrina Parvin. I have completed my Graduation in Mathematics, Having 10 years of experience working in the Ministry of communication and information technology of India. In my free time, I love to teach and solve math problems. From my childhood, Math is the only subject that fascinated me the most.

Basic Examples on the Formulae “Point sections or Ratio”

Hi Fellow Reader,