Which rule describes the composition of transformations that maps rectangle pqrs to p''q''r''s''? r0,270° ∘ t0,2(x, y) r0,180° ∘ t2,0 (x, y) t0,2 ∘ r0,270°(x, y) r0,2 ∘ t0,180°(x, y)

 The correct option is

(C) T0,2(x, y)∘ R0, 270°.

Step-by-step explanation:  We are given to select the rule that describes the composition of transformations that maps rectangle PQRS to P''Q''R''S''.

From the graph, we note that

the co-ordinates of the vertices of rectangle PQRS are P(-3, -5), Q(-2, -5), R(-2, -1) and S(-3, -1).

and the co-ordinates of the vertices of rectangle P''Q''R''S'' are P''(-5, 5), Q''(-5, 4), R''(-1, 4) and S(-1, 5).

Since the rectangle PQRS lies in the Quadrant III and P''Q''R''S'' lies in Quadrant II, so the rotation will be either 90° clockwise or 270° counterclockwise about the origin.

In the given options, we do not have rotation of 90°, so will consider the rotation through 270° counterclockwise about the origin.

Now, after this rotation, the vertices will transform according to the rule  

(x, y)   ⇒   (y, -x).

Therefore, the vertices of the image rectangle P'Q'R'S', after rotation through  270° counterclockwise about the origin, becomes

P(-3, -5)    ⇒  P'(-5, 3),

Q(-2, -5)   ⇒  Q'(-5, 2),

R(-2, -1)     ⇒  R'(-1, 2),

S(-3, -1)     ⇒ S'(-1, 3).

Now, to make the vertices of P'Q'R'S' coincide with the vertices of P''Q''R"S", we need to add 2 units to the y co-ordinate of each vertex , so that

P'(-5, 3)    ⇒   P''(-5, 3+2) = P''(-5, 5),

Q'(-5, 2)   ⇒   Q"(-5, 2+2) = Q''(-5, 4),

R'(-1, 2)     ⇒   R''(-1, 2+2) = R"(-1, 4),

S'(-1, 3)     ⇒   S''(-1, 3+2) = S''(-1, 5).

Thus, the required transformation rule is

rotation through 270° counterclockwise about the origin and a translation of (x, y)   ⇒  (x, y+2).

Since the rigid transformations are written from right to left, so option (C) is correct.

third option: 6x+8

step-by-step explanation:

we can see that the given question is a sum of series and the values of n vary from 0 to 3

the first term will be:

when n=0, [(0)*x+2] => (0+2) => 2

when n=1, [(1)*x+2] => (x+2) => x+2

when n=2, [(2)*x+2] => 2x+2

when n=4, [(3)*x+2] => 3x+2

so adding up all the terms will give us:

=2+x+2+2x+2+3x+2

=x+2x+3x+2+2+2+2

=6x+8

so,

third option is the correct answer ..

4 and 5)16.97 or 12√2

6)15.59 or 9√3

7)y = 15.59 or 9√3     x =27

8)17.80