**Transformation III, Long Questions (Question 2)**

**Question 3:**

**(a)**Transformation

**P**is a reflection in the line

*x*=

*m*.

Transformation

**T**is a translation $\left(\begin{array}{l}\text{4}\\ -2\end{array}\right)$ .Transformation

**R**is a clockwise rotation of 90^{o}about the centre (0, 4).(i) The point (6, 4) is the image of the point ( –2, 4) under the transformation

**P**.

State the value of

*m*.(ii) Find the coordinates of the image of point (2, 8) under the following combined transformations:

(a)

**T**,^{2}(b)

**TR**.

**(b)**Diagram below shows trapezium

*CDFE*and trapezium

*HEFG*drawn on a Cartesian plane.

(i)

*HEFG*is the image of*CDEF*under the combined transformation**WU**.Describe in full the transformation:

(a)

**U**(b)**W**(ii) It is given that

*CDEF*represents a region of area 60 m

^{2}.

Calculate the area, in m

^{2}, of the region represented by the shaded region.

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

$\begin{array}{l}\left(6,4\right)\to P\to \left(-2,4\right)\\ m=\frac{6+\left(-2\right)}{2}=2\end{array}$

**(a)(ii)**

**(a)**(2, 8) →

**T**→ (6, 6) →

**T**→ (10, 4)

**(b)**(2, 8) →

**R**→ (4, 2) →

**T**→ (8, 0)

(b)(i)(a)

(b)(i)(a)

**U**: An anticlockwise rotation of 90

^{o}about the centre

*A*(3, 3).

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

$\text{Scale factor}=\frac{HE}{CD}=\frac{4}{2}=2$

**W**: An enlargement of scale factor 2 with centre*B*(3, 5).

**(b)(ii)**

Area of

*HEFG*= (Scale factor)^{2}× Area of object = 2

^{2}× area of*CDEF* = 4 × 60

= 240 m

^{2}Therefore,

Area of the shaded region

= Area of

*HEFG*– area of*CDEF*= 240 – 60

=

**180 m**^{2}

**Question 4:**Diagram below shows three triangles

*RPQ*,

*UST*and

*RVQ*, drawn on a Cartesian plane.

**(a)**Transformation

**R**is a rotation of 90

^{o}, clockwise about the centre

*O*.

Transformation

**T**is a translation $\left(\begin{array}{l}2\\ \text{3}\end{array}\right)$ .

State the coordinates of the image of point B under each of the following transformations:

(i) Translation

**T**,

^{2}(ii) Combined transformation

**TR**.

**(b)**

**(i)**Triangle

*UST*is the image of triangle

*RPQ*under the combined transformation

**VW**.

Describe in full the transformation:

(a)

**W**(b)

**V**

(ii) It is given that quadrilateral

*RPQ*represents a region of area 15 m

^{2}.

Calculate the area, in m

^{2}, of the region represented by the shaded region.

*Solution*:**(a)**

**(i)**(–5, 3) →

**T**→ (–3, 6) ) →

**T**→ (–1, 9)

**(ii)**(–5, 3) →

**R**→ (3, 5) →

**T**→ (5, 8)

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

**W**: A reflection in the line

*URQT.*

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

$\begin{array}{l}\text{Scalefactor}=\frac{US}{RV}=\frac{6}{2}=3\\ \text{V:Anenlargementofscalefactor3atcentre}\left(-4,2\right)\end{array}$

**(b)(ii)**

Area of

*UST*= (Scale factor)

^{2}x Area of

*RPQ*

= 3

^{2}x area of

*RPQ*

= 3

^{2}x 15

= 135 m

^{2}

Therefore,

Area of the shaded region

= Area of

*UST*– area of

*RPQ*

= 135 – 15

=

**120 m**

^{2}