Problem Set 5 Co-ordinate Geometry Class 10th Mathematics Part 2 MHB Solution

Problem Set 5

  1. Seg AB is parallel to Y-axis and coordinates of point A are (1,3) then co-ordinates of…
  2. Out of the following, point ........ lies to the right of the origin on X- axis. Fill…
  3. Distance of point (-3,4) from the origin is ...... . Fill in the blanks using correct…
  4. A line makes an angle of 30° with the positive direction of X- axis. So the slope of…
  5. Determine whether the given points are collinear. (1) A(0,2) , B(1,-0.5), C(2,-3) (2)…
  6. Find the coordinates of the midpoint of the line segment joining P(0,6)and Q(12,20).…
  7. Find the ratio in which the line segment joining the points A(3,8) and B(-9, 3)is…
  8. Find the point on X-axis which is equidistant from P(2,-5) and Q(-2,9).…
  9. Find the distances between the following points. (i) A(a, 0), B(0, a) (ii) P(-6, -3),…
  10. Find the coordinates of the circum centre of a triangle whose vertices are (-3,1),(0,-2)…
  11. In the following examples, can the segment joining the given points form a triangle? If…
  12. Find k if the line passing through points P(-12,-3) and Q(4, k)has slope 1/2 .…
  13. Show that the line joining the points A(4, 8) and B(5, 5) is parallel to the…
  14. Show that points P(1,-2), Q(5,2), R(3,-1), S(-1,-5) are the vertices of…
  15. Show that the □PQRS formed by P(2,1), Q(-1,3), R(-5,-3) and S(-2,-5) is a rectangle.…
  16. Find the lengths of the medians of a triangle whose vertices are A(-1,1), B(5, -3) and…
  17. Find the coordinates of centroid of the triangles if points D(-7,6),E(8,5) andF(2, -2)…
  18. Show that A(4, -1), B(6, 0), C(7, -2) and D(5, -3)are vertices of a square.…
  19. Find the coordinates of circumcentre and radius of circumcircle of triangle ABC ifA(7,…
  20. Given A(4,-3), B(8,5). Find the coordinates of the point that divides segmentAB in the…
  21. Find the type of the quadrilateral if points A(-4, -2), B(-3, -7) C(3, -2) andD(2, 3)…
  22. The line segment AB is divided into five congruent parts at P, Q, R and S suchthat…
  23. Find the coordinates of the centre of the circle passing through the points P(6,-6),…
  24. Find the possible pairs of coordinates of the fourth vertex D of the parallelogram,if…
  25. Find the slope of the diagonals of a quadrilateral with verticesA(1,7), B(6,3),…

Problem Set 5
Question 1.

Fill in the blanks using correct alternatives.

Seg AB is parallel to Y-axis and coordinates of point A are (1,3) then co-ordinates of point B can be ........ .
A. (3,1)

B. (5,3)

C. (3,0)

D. (1,-3)


Answer:

To be parallel to y-axis, it's x coordinate should remain the same. i.e. 1 and y coordinate can change.


A. x coordinate has changed.


B. x coordinate has changed.


C. x coordinate has changed.


D. x coordinate is same.


Therefore the answer is D.


Question 2.

Fill in the blanks using correct alternatives.

Out of the following, point ........ lies to the right of the origin on X- axis.
A. (-2,0)

B. (0,2)

C. (2,3)

D. (2,0)


Answer:

To be on the X-axis, it's y coordinate = 0


And to be on the right of the origin, its x coordinate must be positive.


A. y is 0 but x is negative


B. y is not 0


C. y is not 0


D. y is 0 and x is positive.


Therefore the answer is D


Question 3.

Fill in the blanks using correct alternatives.

Distance of point (-3,4) from the origin is ...... .
A. 7

B. 1

C. 5

D. -5


Answer:

According to the distance formula, the distance 'd' between two points (a,b) and (c,d) is given by
 .....(1)


d =  = 5


Therefore answer is C


Question 4.

Fill in the blanks using correct alternatives.

A line makes an angle of 30° with the positive direction of X- axis. So theslope of the line is .......... .
A. 

B. 

C. 

D. 


Answer:

Slope = tangent of angle formed with positive x-axis


Slope = tan30° = 


Hence answer is C


Question 5.

Determine whether the given points are collinear.

(1) A(0,2) , B(1,-0.5), C(2,-3)

(2) P(1, 2) , Q(2, 8/5) , R(3, 6/5)

(3) L(1,2) , M(5,3) , N(8,6)


Answer:

If Three points (a,b), (c,d), (e,f) are collinear then the area formed by the triangle by the three points is zero.


Area of a triangle = ...(1)


1. area = 


Hence the points are collinear.


2. area = 


Hence the points are collinear


3. area = 


Hence the points are not collinear



Question 6.

Find the coordinates of the midpoint of the line segment joining P(0,6)and Q(12,20).


Answer:

According to the mid point theorem the coordinates of the point P(x,y) dividing the line formed by A(a,b) and B(c,d) is given by:




In question 



Hence mid point is (6,13)



Question 7.

Find the ratio in which the line segment joining the points A(3,8) and B(-9, 3)is divided by the Y-axis.


Answer:

A point P(x,y) divides the line formed by points (a,b) and (c,d) in the ratio of m:n, then the coordinates of the point P is given by


 and 


On y axis x coordinate = 0


Let the y-axis divide AB is ratio 1:k.

the points A(3,8) and B(-9, 3)is divided by the Y-axis.

For x coordinate



Solving for k we get


3k - 9 = 0

3 k = 9


k = 3


Hence the y axis divides the given point in the ratio 3:1


Question 8.

Find the point on X-axis which is equidistant from P(2,-5) and Q(-2,9).


Answer:

According to the distance formula, the distance 'd' between two points (a,b) and (c,d) is given by
 .....(1)


As the point is on the x = axis it is of the form (x,0)


Distance from point P =  = 


Distance from point Q =  = 


As the two points are equidistant from (x,0)



Squaring both sides


(2-x)2 + 25 = (2 + x)2 + 81


Expanding and simplifying


-4x + 25 = 4x + 81


8x = -56


x = -7


Hence the point is (-7,0)


Question 9.

Find the distances between the following points.

(i) A(a, 0), B(0, a)

(ii) P(-6, -3), Q(-1, 9)

(iii) R(-3a, a), S(a, -2a)


Answer:

According to the distance formula, the distance 'd' between two points (a,b) and (c,d) is given by
 .....(1)



(i) A(a, 0), B(0, a)

i. d = 




(ii) P(-6, -3), Q(-1, 9)

d = 




= √169

= 13


(iii) R(-3a, a), S(a, -2a)

d = 




= 5a


Question 10.

Find the coordinates of the circumcentre of a triangle whose vertices are (-3,1),(0,-2) and (1,3)


Answer:

The circumcentre is equidistant from all the points of the triangle.


Let the coordinates of circumcentre be (x,y)


 =  ...i


And


 ...ii


Squaring and simplifying i , we get


2x-2y = -6


Squaring and simplifying ii , we get


2x + 10y = 6


Solving the above equations, we get


x = 


y = 


hence the coordinates of circumcircle is ()



Question 11.

In the following examples, can the segment joining the given points form atriangle? If triangle is formed, state the type of the triangle considering sides ofthe triangle.

(1) L(6,4), M(-5,-3) , N(-6,8)

(2) P(-2,-6), Q(-4,-2), R(-5,0)

(3) 


Answer:

According to the distance formula, the distance 'd' between two points (a,b) and (c,d) is given by
 .....(1)


1. LM =  = 


MN =  = 


NL =  = 


As sum of any two sides are greater than the third side,


The following points form a scalene triangle.


2. PQ =  = 


QR =  = 


RP =  = 


As PQ + QR < RP


The following points donot form a triangle.


3. AB =  = 4


BC = =  = 4


AC =  = 4


As AB = BC = AC


The following points form an equilateral triangle.



Question 12.

Find k if the line passing through points P(-12,-3) and Q(4, k)has slope .


Answer:

Slope of a line between two points (a,b) and (c,d) is



Slope =  = 


Simplifying


K = 5



Question 13.

Show that the line joining the points A(4, 8) and B(5, 5) is parallel to the linejoining the points C(2,4) and D(1,7).


Answer:

Slope of a line between two points (a,b) and (c,d) is



Slope of AB = 


Slope of AC = 


As slopes are equal, the two lines are parallel.



Question 14.

Show that points P(1,-2), Q(5,2), R(3,-1), S(-1,-5) are the vertices of aparallelogram.


Answer:

According to the distance formula, the distance 'd' between two points (a,b) and (c,d) is given by
 .....(1)


Slope of a line between two points (a,b) and (c,d) is



Distance PQ = 


Distance QR =  = 


Distance RS =  = 


Distance SP =  = 


Slope PQ = 


Slope QR = 


Slope RS = 


Slope SP = 


As opposite sides are equal and parallel, the points from a parallelogram.



Question 15.

Show that the □PQRS formed by P(2,1), Q(-1,3), R(-5,-3) and S(-2,-5) is a rectangle.


Answer:

According to the distance formula, the distance 'd' between two points (a,b) and (c,d) is given by
 .....(1)


Slope of a line between two points (a,b) and (c,d) is



Note: If the Product of slopes of two lines = -1 then they are perpendicular to each other.


PQ =  = 


QR =  = 


RS =  = 


SP =  = 


Slope PQ = 


Slope QR = 


Slope RS = 


Slope SP = 


As opposite sides are equal and parallel and perpendicular to each other, points form a rectangle.



Question 16.

Find the lengths of the medians of a triangle whose vertices are A(-1,1), B(5, -3) and C(3, 5).


Answer:

According to the distance formula, the distance 'd' between two points (a,b) and (c,d) is given by  .....(1)


According to the mid point theorem the coordinates of the point P(x,y) dividing the line formed by A(a,b) and B(c,d) is given by:




Mid point of AB x coordinate = 


Y coordinate = 


Mid point of BC x coordinate = 


Y coordinate = 


Mid point of AC x coordinate = 


Y coordinate =  = 3


Length of median through A is the distance between pt A and the mid point of BC


Da =  = 5


Length of median through B is the distance between pt B and the mid point of AC


Db =  = 2


Length of median through C is the distance between pt C and the mid point of AB


Dc =  = 



Question 17.

Find the coordinates of centroid of the triangles if points D(-7,6),E(8,5) andF(2, -2) are the mid points of the sides of that triangle.


Answer:

The coordinates of the centroid (x,y) od a triangle formed by points (a,b), (c,d), (e,f) is given by




X coordinate = 


Y coordinate = 


Hence coordinates are (1,3)



Question 18.

Show that A(4, -1), B(6, 0), C(7, -2) and D(5, -3)are vertices of a square.


Answer:

According to the distance formula, the distance 'd' between two points (a,b) and (c,d) is given by
 .....(1)


Slope of a line between two points (a,b) and (c,d) is



Note: If the Product of slopes of two lines = -1 then they are perpendicular to each other.


AB =  = 


BC = 


CD =  = 


AD =  = 


Slope AB = 


Slope BC = 


Slope CD = 


Slope AD = 


As all sides are equal and ajdacent sides are perndicular. Given points form a square.



Question 19.

Find the coordinates of circumcentre and radius of circumcircle of triangle ABC ifA(7, 1), B(3, 5) and C(2, 0) are given.


Answer:

Let the circumcentre be (x,y)


As the circumcentre is equidistant from all the 3 points, we get


 =  .......i


And


....................ii


Squaring both sides of i and simplifying, we get


-2x-10y = -30


Squaring both sides of ii and simplifying, we get


10x + 2y = 46


Solving the above equations, we get


x = 


y = 


Radius of circumcircle is the distance between any point on the triangle and the circumcentre.


Radius =  = 



Question 20.

Given A(4,-3), B(8,5). Find the coordinates of the point that divides segmentAB in the ratio 3 : 1.


Answer:

A point P(x,y) divides the line formed by points (a,b) and (c,d) in the ratio of m:n, then the coordinates of the point P is given by


 and 


X coordinate =  = 7


Y coordinate =  = 3


Hence the point is (7,3)



Question 21.

Find the type of the quadrilateral if points A(-4, -2), B(-3, -7) C(3, -2) andD(2, 3) are joined serially.


Answer:

According to the distance formula, the distance 'd' between two points (a,b) and (c,d) is given by
 .....(1)


Slope of a line between two points (a,b) and (c,d) is



AB =  = 


BC =  = 


CD =  = 


AD =  = 


Slope AB = 


Slope BC = 


Slope CD = 


Slope AD = 


As opposite sides are equal and parallel, it forms a parallelogram.



Question 22.

The line segment AB is divided into five congruent parts at P, Q, R and S suchthat A-P-Q-R-S-B. If point Q(12, 14) and
S(4, 18) are given find thecoordinates of A, P, R,B.


Answer:

A point P(x,y) divides the line formed by points (a,b) and (c,d) in the ratio of m:n, then the coordinates of the point P is given by


 and 


Coordinates of R as QR:RS ∷ 1:1


X = 


Y =  = 16


As RS:SB∷1:1


Coordinates of B



X = 0


And



Y = 20


As PQ:QR∷1:1


Coordinates of P



x = 16


and



y = 12


As AP:PQ∷1:1


Coodinates of A



x = 20


and



y = 10



Question 23.

Find the coordinates of the centre of the circle passing through the points P(6,-6), Q(3,-7)and R(3,3).


Answer:

According to the distance formula, the distance 'd' between two points (a,b) and (c,d) is given by
 .....(1)


Let the centre be A(x,y)


As it passes through the given points, distance between centre and the points is the radius.


AP = 


AQ = 


AR = 


As AP = AQ


Squaring both sides


(x-6)2 + (y + 6)2 = (x-3)2 + (y + 7)2


Simplifying


12y-12x + 72 = 14y-6x + 58


2y + 6x-14 = 0... (a)


AP = AR


Squaring both sides and simplifying


6y-6x + 54 = 0 ...(b)


Solving for x and y using (a) and (b)


We get x = 3; y = -2


Hence centre is (3,-2)



Question 24.

Find the possible pairs of coordinates of the fourth vertex D of the parallelogram,if three of its vertices are A(5,6), B(1,-2)and C(3,-2).


Answer:

According to the distance formula, the distance 'd' between two points (a,b) and (c,d) is given by
 .....(1)


Slope of a line between two points (a,b) and (c,d) is



In the given question, for it to be a parallelogram AD = BC and slope AD = Slope BC


And


AB = DC and Slope AB = Slope CD


Let D be (x,y)


As AD = BC we get


 =  = 2 ....i


As AB = CD we get


 =  = 4 .....ii


As slope AD = Slope BC


 ....iii


As slope AB = Slope DC


.....iv


From iii we gey y = 6


Putting y = 6 in (i) we get


x = 3


and putting y = 6 in (ii) we get x = 7


Hence the possible coordinates of the point D are (7,6) and (3,6).



Question 25.

Find the slope of the diagonals of a quadrilateral with verticesA(1,7), B(6,3), C(0,-3) and D(-3,3).


Answer:

Slope of a line between two points (a,b) and (c,d) is



A quadrilateral ABCD has diagonals AC and BD


Slope AC = 


Slope BD = 


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