**Question 13**:

**(a)**Diagram 7.1 shows point K on a Cartesian plane.

**Diagram 7.1**

Diagram 7.1 shows point K on a Cartesian plane.

Transformation **P** is a reflection in the line y = 1.

Transformation R is a clockwise rotation of 90° about the centre (-1,0)

State the coordinates of the image of point K under each of the following transformations:**(i)** **P**,**(ii)** **PR**.

[3 marks]

**(b)**Diagram 7.2 shows three pentagons

*EFGHJ*,

*KLMNP*, and

*KQRST*, drawn on a Cartesian plane.

**Diagram 7.2**

**(i)**Pentagon

*KQRST*is the image of pentagon

*EFGHJ*under the combined transformation

**.**

*VU*Describe, in full, the transformation:

**(a)**

**U**,

**(b) V**.

**(ii)**It is given that the pentagon

*KQRST*represents a region of area 288 m

^{2}.

Calculate the area, in m

^{2}, of the shaded region.

[9 marks].

**Solution**:**(a)**

**(a)(i)**

Point

*K*(3, –2) is reflected in the line

*y*= 1.

*(3, –2) →*

K

K

**P**→

*K*’ (3, 4).

Coordinates of image are (3, 4).

**(a)(ii)**

**(3, –2) →**

*K***R**→

*K*’ (–3, –4) →

**P**→

*K*’’ (–3, 6).

Coordinates of image are (–3, 6).

**(b)**

**(b)(i)(a)**

**U**: 90

^{o}rotation in clockwise direction at centre (–3, 3).

**(b)(i)(a)**

**V**: enlargement at centre

*K*(–1, 8) with scale factor 2.

**(b)(ii)**

Area of

*KQRST*= (scale factor)

^{2}× Area of

*KLMNP*

288 = 2

^{2}× Area of

*KLMNP*

Area of

*KLMNP*= 72 m

^{2}Area of shaded region

= Area of

*KQRST*– Area of

*KLMNP*

= 288 m

^{2 }– 72 m

^{2}= 216 m

^{2}

^{}