3.1 Three-dimensional geometryIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   The points $P(2,-1,4)$ and $Q(8,3,-8)$ lie in three-dimensional space, with all lengths in metres.

   (a)

   Find the distance $PQ$.

   [1 mark]

   • A$196$ m
   • B$22$ m
   • C$\sqrt{120}$ m
   • D$14$ m

   (b)

   Find the coordinates of the midpoint of $[PQ]$.

   [1 mark]

   • A$(10,2,-4)$
   • B$(5,1,-2)$
   • C$(3,2,-6)$
   • D$(6,4,-12)$

   (c)

   The point $R$ is such that $Q$ is the midpoint of $[PR]$. Find the coordinates of $R$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A toy is made from a solid hemisphere of radius 6 cm and a solid cone of base radius 6 cm and vertical height 8 cm. The base of the cone is fixed exactly onto the flat circular face of the hemisphere.

   (a)

   Find the volume of the toy.

   [1 mark]

   • A$829$ cm$^3$
   • B$1206$ cm$^3$
   • C$754$ cm$^3$
   • D$1357$ cm$^3$

   (b)

   Find the total area of the curved surfaces of the toy.

   [1 mark]

   • A$415$ cm$^2$
   • B$377$ cm$^2$
   • C$528$ cm$^2$
   • D$641$ cm$^2$

   (c)

   Find the angle between the slant height of the cone and its circular base.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A tent is in the shape of a right pyramid with square base $ABCD$ of side 6 m. The vertex $V$ is vertically above the centre $M$ of the base, and $VM=4$ m.

   (a)

   Find the length of the edge $VA$.

   [3 marks]

   (b)

   (i) Find the angle between the edge $VA$ and the base $ABCD$.
   (ii) Find the volume of the tent.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A box is a cuboid $ABCDEFGH$ with horizontal base $ABCD$, where $E$, $F$, $G$ and $H$ are directly above $A$, $B$, $C$ and $D$ respectively. $AB=12$ cm, $BC=5$ cm and $AE=8$ cm. Take $A$ as the origin, with $B$ on the $x$-axis, $D$ on the $y$-axis and $E$ on the $z$-axis, with units in cm.

   (a)

   (i) Find the length of $AC$.
   (ii) Find the length of $AG$.
   (iii) Find the angle between $AG$ and the base $ABCD$.

   [6 marks]

   (b)

   Let $M$ be the midpoint of $[AG]$.
   (i) Find the coordinates of $M$.
   (ii) Find the distance $MB$.
   (iii) Show that $M$ is also the midpoint of $[BH]$.

   [6 marks]

   Total for question 4: 12 marks

End of questions