21M.1.SL.TZ2.2

pestleMathematicsAISLPaper 121M· sl-3-1-3d-space-volume-angles-midpointssource ↗

An inclined railway travels along a straight track on a steep hill, as shown in the diagram.

The locations of the stations on the railway can be described by coordinates in reference to $x, y$ , and $z$ -axes, where the $x$ and $y$ axes are in the horizontal plane and the $z$ -axis is vertical.

The ground level station $A$ has coordinates $(140, 15, 0)$ and station $B$ , located near the top of the hill, has coordinates $(20, 5, 250)$ . All coordinates are given in metres.

Station $M$ is to be built halfway between stations $A$ and $B$ .

Find the distance between stations $A$ and $B$ .

[2]

Find the coordinates of station $M$ .

[2]

Write down the height of station $M$ , in metres, above the ground.

[1]

Markscheme / solution

attempt at substitution into 3D distance formula (M1)

$AB = \sqrt{{(140 - 20)}^{2} + {(15 - 5)}^{2} + 250^{2}} (= \sqrt{77 000})$

$= 277 m (10 \sqrt{770}, 277.488 \dots)$ A1

[2 marks]

attempt at substitution in the midpoint formula (M1)

$(\frac{140 + 20}{2}, \frac{15 + 5}{2}, \frac{0 + 250}{2})$

$(80, 10, 125)$ A1

[2 marks]

$125 m$ A1

[1 mark]

Examiners’ report

[N/A]