The vertices of a ∆ ABC are A(4, 6), B(1, 5) and C(7, 2). A line is drawn to intersect sides AB and AC at D and E respectively so that [AD/AB] =[AE/AC] =[1/4] Calculate the area of ∆ ADE and compare it with the area of ∆ ABC

 

exercise 7.4 optional exercise co ordinate geometry chapter 7 cbse ncert 10th mathematics

6. The vertices of a ∆ ABC are A(4, 6), B(1, 5) and C(7, 2). A line is drawn to intersect sides AB and AC at D and E respectively so that [AD/AB] =[AE/AC] =[1/4] Calculate the area of  ∆ ADE and compare it with the area of ∆ ABC

[AD/AB] =[AE/AC] =[1/4]

means

 

AB =4AD  and AC=4AE

 [ draw a line AB and plot D subject to AB =4AD ]

This shows that 

AD / DB=1/3 so that D divides AB internally in the ratio 1:3

similarly E divides AC in the ratio 1:3 internally.

 

 A(4, 6), B(1, 5) ratio =1:3

using section formula

 

D = (  [1*1 + 3*4 ]/[1+3]  , [1*5+3*6 ]/[1+3]  )

D = ( 13/4 , 23/4 )

A(4, 6), C(7, 2) ratio =1:3

E = ( [ 1*7 +  3*4 ]/ [1+3] , [ 1*2 +3*6]/[1+3] )

E = (19/4 , 5 )

  A(4, 6), B(1, 5) and C(7, 2)

 

area of triangle ABC =(1/2)[4(5-2) +1(2-6)+7(6-5)]

=(1/2)[12+(-4)+7]=(15/2)

 

A(4, 6), D ( 13/4 , 23/4) , E = (19/4 , 5 )

 

area of triangle ADE =(1/2) [ 4( (23/4) -5 ) +(13/4)( 5 - 6 )+(19/4)( 6- (23/4)) ]

=(1/2) [ 4(3/4) +(13/4)(-1) +(19/4)(1/4) ]

=(1/2) [15/16] =[15/32]

[area of  ADE] / [area of ABC] =[15/32] / [15/2] = 1/16

ratio  = 1:16

7. Let A (4, 2), B(6, 5) and C(1, 4) be the vertices of ∆ ABC.
(i) The median from A meets BC at D. Find the coordinates of the point D.

(ii) Find the coordinates of the point P on AD such that AP : PD = 2 : 1

 

(i) 

D is the midpoint of BC

D = ( [6+1]/2  , [5+4]/2 )

D= (7/2 , 9/2) 

 

(ii)

A = (4, 2),

D= (7/2 , 9/2) 

ratio = 2:1

 

P = (  [ 2*(7/2) + 1*4 ] / [2+1]  , [ 2(9/2) +1*2]/[2+1] )

P=( [11/3] , [11/3])

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ncert cbse 10th mathematics

 

co ordinate geometry chapter 7

exercise 7.4 optional exercise  

 

Determine the ratio in which the line 2x + y – 4 = 0 divides the line segment joining the points A(2, – 2) and B(3, 7).

 solution 

 

2. Find a relation between x and y if the points (x, y), (1, 2) and (7, 0) are collinear.

solution 

 

3. Find the centre of a circle passing through the points (6, – 6), (3, – 7) and (3, 3).

solution

4. The two opposite vertices of a square are (–1, 2) and (3, 2). Find the coordinates of the other two vertices.

solution   

6. The vertices of a ∆ ABC are A(4, 6), B(1, 5) and C(7, 2). A line is drawn to intersect sides AB and AC at D and E respectively so that [AD/AB] =[AE/AC] =[1/4] Calculate the area of  ∆ ADE and compare it with the area of ∆ ABC

solution

7. Let A (4, 2), B(6, 5) and C(1, 4) be the vertices of ∆ ABC.
(i) The median from A meets BC at D. Find the coordinates of the point D.

(ii) Find the coordinates of the point P on AD such that AP : PD = 2 : 1  

solution 

 

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