The vertices of polygon abcd are at a(1, 1), b(2, 3), c(3, 2), and d(2, 1). abcd is reflected across the x-axis and translated 2 units up to form polygon a′b′c′d′. match each vertex of polygon a′b′c′d′ to its coordinates.
tiles
(2, 1)
a′
(1, 1)
b′
(3, 0)
c′
(2, 3)
d′
(-3, 4)
(2, -1)
(-2, 5)

 The required match is given by

A'         (1, 1)

B'         (2, -1)

C'         (3, 0)

D'         (2, 1).

Step-by-step explanation:  Given that the vertices of polygon ABCD are at A(1, 1), B(2, 3), C(3, 2), and D(2, 1).

ABCD is reflected across the x-axis and translated 2 units up to form polygon A′B′C′D′.

We are to match the vertices of polygon ABCD to its co-ordinates.

We know that

if a point (x, y) is reflected across X-axis, hen the sign before the y co-ordinate changes. Also, if there is an additional translation of 2 units up, then the required transformation will be

(x, y)  ⇒  (x, -y + 2).

So, after getting reflected across the X-axis, the co-ordinates of the vertices of ABCD will change as follows :

A(1, 1)  ⇒  A'(1, -1+2) = A'(1, 1)

B(2, 3)  ⇒ B'(2, -3+2) = B'(2, -1)

C(3, 2)  ⇒  C'(3, -2+2) = C'(3, 0)

and

D(2, 1)  ⇒  D'(2, -1+2) = D'(2, 1).

Thus, the required match is given by

A'         (1, 1)

B'         (2, -1)

C'         (3, 0)

D'         (2, 1).

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substitute,

circle 1 = 5

circle 2 = 20

circle 3 = 44

in this case no matter how you add, you will always get a perfect square..

hence proved

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