Coordinate Geometry Ex- 7.2 ( Section Formula )

 Hi Friends and Champs!

Before we proceed to next level, lets' understand , what is a section formula of a point P(x,y) which divides a line AB with coordinates A (x₁ ,y₁) and B (x₂ ,y₂) in the ratio m₁: m₂

Now, how do we get this formula, understand this:

Draw a line AB in XY plane and P(x,y) is point at line AB . Let's take coordinates of line AB are A (x₁ ,y₁) and B (x₂ ,y₂). 

Construction :

1. Draw AE and  PC parallel to x- axis.

2. Draw AF, PG and BH perpendicular to x - axis.

In Δ APD and Δ PBC 

ΔAPD ~ Δ PBC   ( AA criterion of similarity )

so AP/PB = AD/PC = PD/BC

AP = m₁

PB = m₂

AD = FG = √ [(x _- x₁)² + (0 - 0)²] = x _- x₁

PC = GH = √ [(x_{2 -} x)² + (0 - 0)²] = x_{2 -} x

PD =  √ [(x _- x)² + (y - y₁)²] = y - y₁

PD =  √ [(x_2 - x₂)² + (y₂ - y )²] = y₂ - y

 

=> m_1/m_2 = (x- x₁) / (x_{2 -} x) = (y _- y₁) / (y_2 - y)

=> m₁(x_{2 -} x) ₌ m₂(x- x₁)

=> m₁x_{2 -} m₁x ₌ m₂x- m₂x₁

=> x = ( m₁x₂+ m₂x₁) / ( m₁+ m₂)

similarly

 y = ( m₁y₂+ m₂y₁) / ( m₁+ m₂)

Hence section formula for point P is:

If  m_{1 =} m₂ i. e. P is the mid point of line AB , then

This is called Mid-Point section formula.

                                                

                                                                Exercise 7.2

 1. Find the coordinates of the point which divides the join of (–1, 7) and (4, –3) in the ratio 2 : 3. 

Solution:

 2. Find the coordinates of the points of trisection of the line segment joining (4, –1) and (–2, –3).

Solution:

3. To conduct Sports Day activities, in your rectangular shaped school ground ABCD, lines have been drawn with chalk powder at a distance of 1m each. 100 flower pots have been placed at a distance of 1m from each other along AD, as shown in Fig. 7.12. Niharika runs 1/4 distance AD on the 2nd line and posts a green flag. Preet runs 1/5 th the distance AD on the eighth line and posts a red flag. What is the distance between both the flags? If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, where should she post her flag? 

Solution:

4. Find the ratio in which the line segment joining the points (– 3, 10) and (6, – 8) is divided by (– 1, 6).

Solution:

 5. Find the ratio in which the line segment joining A(1, – 5) and B(– 4, 5) is divided by the x-axis. Also find the coordinates of the point of division. 

Solution:

 6. If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y. 

Solution :

 7. Find the coordinates of a point A, where AB is the diameter of a circle whose Centre is (2, – 3) and B is (1, 4).

Solution:

8. If A and B are (– 2, – 2) and (2, – 4), respectively, find the coordinates of P such that Fig. 7.12 AP = 3/ 7 AB and P lies on the line segment AB.

Solution:

9. Find the coordinates of the points which divide the line segment joining A(– 2, 2) and B(2, 8) into four equal parts. 

Solution:

10. Find the area of a rhombus if its vertices are (3, 0), (4, 5), (– 1, 4) and (– 2, – 1) taken in order. [Hint : Area of a rhombus = 1/2 (product of its diagonals)]

Solution:

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