Question

[Maths]
Topic:- Reflection (Co-ordinate Geometry)

The points A(4, -11), B(5, 3) C(2, 15) and D(1, 1) are the vertices of a parallelogram. If the parallelogram is reflected in the y-axis and then in the origin, find the coordinates of the final images. Check whether it remains a parallelogram. Write down a single transformation that brings the above change.
Graph should also be attached.​

Answer 1

Topic :-

Coordinate Geometry

Given :-

The points A (4, -11), B (5, 3), C (2, 15) and D (1, 1) are the vertices of a parallelogram.

To Find :-

If the parallelogram is reflected in the Y-axis and then in the origin, find the coordinates of the final images and check whether it remains a parallelogram or not.

Solution :-

If a point (a, b) is reflected in the Y-axis then its coordinates will be (-a, b) and if point (-a, b) will be reflected in the origin then its coordinates will be (a, -b).

Using above concept, coordinates of final image of vertices of parallelogram after reflecting it in Y-axis and origin will be :-

A' (4, 11)

B' (5, -3)

C' (2, -15)

D' (1, -1)

Calculating distances between these points using Distance formula :-

$\sf{A'B'=\sqrt{(5-4)^2+(-3-11)^2}}$

$\sf{A'B'=\sqrt{1^2+(-14)^2}}$

$\sf{A'B'=\sqrt{1+196}}$

$\sf{A'B'=\sqrt{197}\:units}$

$\sf{B'C'=\sqrt{(2-5)^2+(-15-(-3))^2}}$

$\sf{B'C'=\sqrt{(-3)^2+(-15+3)^2}}$

$\sf{B'C'=\sqrt{9+(-12)^2}}$

$\sf{B'C'=\sqrt{9+144}}$

$\sf{B'C'=\sqrt{153}\:units}$

$\sf{C'D'=\sqrt{(1-2)^2+(-1-(-15))^2}}$

$\sf{C'D'=\sqrt{(-1)^2+(-1+15)^2}}$

$\sf{C'D'=\sqrt{(-1)^2+(14)^2}}$

$\sf{C'D'=\sqrt{1+196}}$

$\sf{C'D'=\sqrt{197}\:units}$

$\sf{D'A'=\sqrt{(4-1)^2+(11-(-1))^2}}$

$\sf{D'A'=\sqrt{(3)^2+(11+1)^2}}$

$\sf{D'A'=\sqrt{(3)^2+(12)^2}}$

$\sf{D'A'=\sqrt{9+144}}$

$\sf{D'A'=\sqrt{153}\:units}$

$\sf{A'C'=\sqrt{(2-4)^2+(-15-11)^2}}$

$\sf{A'C'=\sqrt{(-2)^2+(-26)^2}}$

$\sf{A'C'=\sqrt{4+676}}$

$\sf{A'C'=\sqrt{680}\:units}$

$\sf{B'D'=\sqrt{(1-5)^2+(-1-(-3))^2}}$

$\sf{B'D'=\sqrt{(-4)^2+(-1+3)^2}}$

$\sf{B'D'=\sqrt{(-4)^2+(2)^2}}$

$\sf{B'D'=\sqrt{16+4}}$

$\sf{B'D'=\sqrt{20}\:units}$

From above data we can say that,

A'B' = C'D', B'C' = D'A' and A'C' ≠ B'D' which is the condition for a quadrilateral to be a parallelogram. Hence, images of vertices formed after reflection in Y-axis and origin forms a parallelogram.

Answer :-

Coordinates of final images are :-

A' (4, 11)

B' (5, -3)

C' (2, -15)

D' (1, -1)

After reflection of vertices of parallelogram in Y-axis and origin it still remains a parallelogram.

Note : Check attachment for graphs. First graph is of original parallelogram and second graph is of parallelogram formed after performing given operations.