Cartesion System - Test Questions-ii

Question 1

Question:   In a parallelogram ABCD, the Co-ordinates of the vertices are A(-8, -4), B(6, -4), C(x, y), D(-3, 3). Find the Coordinates of the point C by applying mid point formula. [3 Mark]

Answer:   

The diagonals in a parallelogram bisect each other. Let the point of bisection be O.

Then by mid point formula,

















Coordinates of C(11, 3).

Question 2

Question:   Points A and B have vertices (7, -2) and (a, b) respectively. The Coordinates of the mid point are (4, -3). Find the values of a and b. [2 Mark]

Answer:   

By mid point formula, Coordinates of mid point can be calculated by,











Hence a = 1, b = -4

Coordinates of B(1, -4).

Question 3

Question:   Find the Coordinates of the point A and B where the line 3x - 2y = 24 cuts the x-axis and Y-axis respectively. Also find the Coordinates of the mid point of AB. [3 Mark]

Answer:   

Point A(x, 0) lies on the line 3x - 2y = 24

Hence values of x and y should satisfy the equation,

3x - 2(0) = 24

3x = 24



Coordinates of A (8, 0)

Pt B(0, y) lies on the line 3x - 2y = 24

3(0) - 2y = 24



Co-ordinates of B(0, -12)

Mid point of A(8, 0) , B(0, -12) is given by mid point formula,



= (4, -6)

Hence Co-ordinates of mid points are given by (4, -6).

Question 4

Question:   Show that the points A(-2, 3), B(4, 0) and C(1, -3) are the vertices of an isosceles triangle. Find the Co-ordinates of D so that ABCD is a Rhombus. [3 Mark]

Answer:   

If ABC is an isosceles triangle then AC = AB.

By distance formula







By distance formula







is an isosceles triangle.

If ABCD is a Rhombus, then diagonals bisect each other.

.





O is also the mid point of AD







Coordinates of D (7, -6)

Question 5

Question:   A (-4, -2), B(2, 0), C(8, 6) and D(x, y) are the coordinates of the vertices of a parallelogram. Find the value of x and y. [3 Mark]

Answer:   

The diagonals bisect each other in a parallelogram.

Hence O is the n\mid point of AC and BD.





= (2, 2)









Coordinates of D(2, 4).

Question 6

Question:   A is a point on the positive side of the X-axis and B is a point on the +ve side of Y- axis. P(4, 5) is the mid point of AB. Find the Co-ordinates of A and B. [3 Mark]

Answer:    Given A is a point on +ve side of X-axis

its Coordinates are (x, 0)

B is a point on +ve side of Y-axis

its Co-ordinates are (0, y).

By mid point formula, then mid points



But given the coordinates of mid point = P (4, 5)







Hence the Coordinates of A(8, 0), B(0, 10).

Question 7

Question:   The mid point of the joining (a, 2) and (3, 6) is (2, b). Find the numerical values of a, b. [2 Mark]

Answer:    By mid point formula,











The numerical value of a = 1 and b = 4.

Question 8

Question:   The line segment joining A(2, 3) and (6, -5) is intercepted by x -axis at a point K. Write down the ordinate of the point K. Hence find the ratio in which K divides AB. [3 Mark]

Answer:    The ordinate of point K is 0 as it lies on x-axis by the interception by x-axis of line AB.



Let section formula the Coordinates of K is given by















Hence m:n = 3:5

K divides AB in the ratio 3:5.

Question 9

Question:   Prove that the points A(-5, 4), B(-1, -2) and C(5, 2) are the vertices of an isosceles right angled triangle. Find the Coordinates of D so that ABCD is a square. [3 Mark]

Answer:    If ABC is a right angled triangle, by Pythagoras theorem,



AB² + BC² = AC²

and it is given ABC is also isosceles,

AB = BC, hence,

AB² + AB² = AC²

2AB² = AC²





AB² = 16 + 36 = 52 sq.units

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.



AC² = 10² + 2² = 104 sq. units

AB² + BC² = 52 + 52 = 104 sq. units

AC² = 104 sq. units

Hence AB² + BC² = AC²



If ABCD is a square, then diagonals bisect each other at right angles.

O is the mid point of AC

Also O is the mid point of BD





= (0, 3)

Let Coordinates of D (x, y) then, by mid point formulae,









Co-ordinates of D (1, 8).

Question 10

Question:   ABC is a triangle with vertices A(a, 5), B(6, -2) and C(-7, b). If (4, 2) are the Coordinates of its centroid, find a and b. [3 Mark]

Answer:    Centroid is a point where all the median intersect.



Median is line joining the vertex to the mid point of opposite side. Hence by mid point formula, mid point of AC



and centroid divides the median in the ratio 2:1,

Consider then median from vertex B, By section formula,

Let m = 2, n = 1



Where m = 2, n = 1,







12 = 2x + 6, 6 = 2y - 2

6 = 2x, 8 = 2y

3 = x, 4 = y

Substituting the values of x, y in Co-ordinates of mid point of AC,









Hence a = 13, b = 3