Coordinate geometry and describing movementIB MYP Maths Extended: Subtopic test

1. 1
   $ABCD$ is a rectangle with its sides parallel to the axes on a Cartesian plane. $A$ is $(-2,3)$, $B$ is $(4,3)$ and $C$ is $(4,-1)$. One unit on each axis is $1$ cm.

   (a)

   Find the coordinates of $D$.

   [1 mark]

   • A$(-2,-1)$
   • B$(-1,-2)$
   • C$(2,-1)$
   • D$(-2,1)$

   (b)

   Find the area of the rectangle.

   [1 mark]

   • A$10$ cm$^2$
   • B$20$ cm$^2$
   • C$24$ cm$^2$
   • D$12$ cm$^2$

   (c)

   Describe the translation that maps $A$ onto $C$, in terms of movement right or left and up or down.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Triangle $P$ has vertices $(1,2)$, $(4,2)$ and $(1,6)$ on a Cartesian plane.

   (a)

   $P$ is translated $5$ left and $3$ down. Which point is the image of $(4,2)$?

   [1 mark]

   • A$(9,5)$
   • B$(-1,-1)$
   • C$(-1,5)$
   • D$(9,-1)$

   (b)

   Which movement takes the vertex $(1,6)$ to the point $(6,1)$?

   [1 mark]

   • A$5$ left and $5$ up
   • B$5$ left and $5$ down
   • C$5$ right and $5$ up
   • D$5$ right and $5$ down

   (c)

   Triangle $Q$ has vertices $(6,2)$, $(9,2)$ and $(6,6)$. Describe the movement that maps $P$ onto $Q$, and explain why $P$ and $Q$ are congruent.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Triangle $T$ has vertices $(0,0)$, $(2,0)$ and $(0,3)$. Triangle $U$ has vertices $(0,0)$, $(6,0)$ and $(0,9)$. Triangle $V$ has vertices $(0,0)$, $(4,0)$ and $(0,5)$.

   (a)

   Show that triangle $T$ and triangle $U$ are similar, and state the scale factor of the enlargement from $T$ to $U$.

   [3 marks]

   (b)

   Decide whether triangle $V$ is similar to triangle $T$. Also decide whether triangle $T$ and triangle $U$ are congruent. Give reasons.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A robot vacuum cleaner moves around a room drawn on a Cartesian plane, where one unit is one metre. The room covers $0\le x\le12$ and $0\le y\le10$. The robot starts at $A(1,2)$. A move is described by how far right and how far up the robot goes.

   (a)

   The robot repeats the move '$2$ right, $3$ up' again and again, ignoring the walls for this part.
   (i) Write down its position after $1$, $2$ and $3$ moves.
   (ii) Describe the pattern, and write a rule for its position after $n$ moves.
   (iii) Verify your rule for $n=10$ by a different method.

   [6 marks]

   (b)

   The robot repeats the move '$2$ right, $3$ up' from $A$ until the next move would take it out of the room. Find the number of moves it makes inside the room and its final position. Evaluate how well this model represents a real robot vacuum cleaner.

   [6 marks]

   Total for question 4: 12 marks

End of questions