ABCD is a rectangle formed by the points A(–1, –1), B(– 1, 4), C(5, 4) and D(5, – 1). P, Q, and S are the mid-points of AB, BC, CD and DA respectively. Is the quadrilateral PQRS a square? a rectangle? or a rhombus? Justify your answer.

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

 

8. ABCD is a rectangle formed by the points A(–1, –1), B(– 1, 4), C(5, 4) and

 D(5, – 1). P, Q, and S are the mid-points of AB, BC, CD and DA respectively. Is the quadrilateral PQRS a square? a rectangle? or a rhombus? Justify your answer.

 

  A(–1, –1), B(– 1, 4)

using midpoint formula

P = ( [(-1) +(-1)]/2 , [(-1)+4]/2 ) = (-1 , 3/2 )

 

B(– 1, 4), C(5, 4)

Q = ( [(-1)+5]/2 , [4+4]/2 ) = (2,4)

 

C(5, 4)  D(5, – 1)

R = ( [ 5 + 5]/2 , [4+(-1)]/2 ) = (5, (3/2) )

 

 D(5, – 1) A(–1, –1)

S = ( [5 +  (-1) ]/2  , [(-1) + (-1)] / 2 ) = (2, -1)

 

 

using distance formula

 P = (-1 , 3/2 ) ; Q= (2,4)

PQ = sqrt { (-1 -2)^2 + [(3/2) -4]^2 }  =sqrt[61/4]

Q= (2,4) R=(5, (3/2) )

QR = sqrt{ (2-5)^2 + [4-(3/2)]^2 } =  sqrt[61/4]

 

 R=(5, (3/2) )  S =(2, -1)

RS = sqrt{(5-2)^2 + [(3/2) -(-1)]^2}  = sqrt[61/4]

 

S =(2, -1) P = (-1 , 3/2 )

SP= sqrt{[2-(-1)]^2 + [(-1) - (3/2)]^2 }  =sqrt[61/4]

 

similarly checking diagonals

P = (-1 , 3/2 ) R=(5, (3/2) )

PR = sqrt{16) = 4

 S =(2, -1) Q= (2,4) 

SQ =sqrt{25}  =5

 so diagonals are not equal

 

therefore all sides of the quadrilateral PQRS are equal, but the diagonals are not equal.

so PQRS is a rhombus.

 

exercise 7.3

 Find the area of the triangle whose vertices are

  (2, 3), (–1, 0), (2, – 4)

Area = (1/2) {2 [0-(-4)] +(-1)[(-4)-3] +2[3-0]} 

=(1/2){8+7+6} = (21/2) sq units.

 

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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 

 

8. ABCD is a rectangle formed by the points A(–1, –1), B(– 1, 4), C(5, 4) and

 D(5, – 1). P, Q, and S are the mid-points of AB, BC, CD and DA respectively. Is the quadrilateral PQRS a square? a rectangle? or a rhombus? Justify your answer.

solution

 

exercise 7.3

 Find the area of the triangle whose vertices are

  (2, 3), (–1, 0), (2, – 4)

 solution

 

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