3.5 Perpendicular bisectorsIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   Points $A(2,1)$ and $B(8,5)$ are given.

   (a)

   Find the midpoint of $[AB]$.

   [1 mark]

   • A$(3,2)$
   • B$(6,4)$
   • C$(10,6)$
   • D$(5,3)$

   (b)

   Find the gradient of the perpendicular bisector of $[AB]$.

   [1 mark]

   • A$-\frac32$
   • B$\frac32$
   • C$-\frac23$
   • D$\frac23$

   (c)

   Find the equation of the perpendicular bisector of $[AB]$, giving your answer in the form $ax+by=d$ where $a$, $b$ and $d$ are integers.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Points $P(-3,7)$ and $Q(5,3)$ are given.

   (a)

   Find the midpoint of $[PQ]$.

   [1 mark]

   • A$(4,-2)$
   • B$(1,5)$
   • C$(2,10)$
   • D$(8,-4)$

   (b)

   Which point lies on the perpendicular bisector of $[PQ]$?

   [1 mark]

   • A$(-1,6)$
   • B$(2,3)$
   • C$(2,7)$
   • D$(4,1)$

   (c)

   Find the equation of the perpendicular bisector of $[PQ]$, giving your answer in the form $y=mx+c$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The line segment $[CD]$ lies on the line $3x-4y=10$. The midpoint of $[CD]$ is $M(6,2)$.

   (a)

   Find the equation of the perpendicular bisector of $[CD]$, giving your answer in the form $ax+by=d$ where $a$, $b$ and $d$ are integers.

   [3 marks]

   (b)

   The point $C$ has coordinates $(2,-1)$. Find the coordinates of $D$, and hence find the length of $[CD]$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   On a map where 1 unit represents 1 km, three schools are at $A(1,2)$, $B(9,8)$ and $C(9,-2)$.

   (a)

   (i) Find the midpoint of $[AB]$.
   (ii) Find the equation of the perpendicular bisector of $[AB]$, giving your answer in the form $ax+by=d$ where $a$, $b$ and $d$ are integers.
   (iii) A bus stop on this perpendicular bisector has $y$-coordinate $1$. Find its coordinates.

   [6 marks]

   (b)

   (i) Find the equation of the perpendicular bisector of $[BC]$.
   (ii) Hence find the coordinates of the point that is the same distance from all three schools.
   (iii) Find this distance.

   [6 marks]

   Total for question 4: 12 marks

End of questions