IB Maths AI SL Topic 3: Geometry and trigonometry questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

Paper

Paper 1 Paper 2

Difficulty

Easy Medium Hard

Question 1

Paper 2Medium6 marks

A designer is planning a pavilion with a square-based roof shaped as a right pyramid. The apex $A$ is vertically above the centre $O$ of the square base. Each side of the base is $8.4\text{ m}$ and the current vertical height $AO$ is $4.60\text{ m}$ The design requires the angle between each sloping edge and the plane of the base to be at least $38^\circ$ Let $P$ be a corner of the base.

(a)

Show that $OP=5.94\text{ m}$

[2]

(b)

Calculate the angle between $AP$ and the plane of the base, giving your answer to the nearest $0.1^\circ$

[1]

(c)

Determine the minimum vertical height required to meet the design requirement and hence state whether the current roof meets it.

[3]

Question 2

Paper 2Medium6 marks

Two coastal stations, $A$ and $B$ lie on a straight east–west coastline. Station $B$ is $5.7\text{ km}$ due east of station $A$ A boat at $C$ north of the coastline, is observed on a bearing of $062^\circ$ from $A$ and a bearing of $323^\circ$ from $B$

(a)

Draw a labelled sketch of the situation and determine the interior angles of triangle $ABC$ at $A$ and $B$

[2]

(b)

Determine the distance $AC$ giving your answer to 3 significant figures.

[2]

(c)

Hence, using your answer to part (b), determine the shortest distance from the boat to the coastline. Give your answer to 3 significant figures.

[2]

Question 3

Paper 2Medium6 marks

Three weather stations record annual rainfall. Their coordinates are measured in kilometres.

Station	Coordinates	Annual rainfall (mm)
A	$(1,1)$	540
B	$(7,3)$	680
C	$(3,7)$	610

The Voronoi diagram assigns each location to its nearest station.

(a)

Find the equation of the boundary between the cells for stations A and B, in the form $y=mx+c$

[2]

(b)

The boundary between the cells for stations A and C has equation $x+3y=14$ The boundaries between the cells for A and B, and A and C, meet at a Voronoi vertex. Using your answer to part (a), find the coordinates of this vertex.

[3]

(c)

Estimate the annual rainfall at $Q=(2,5)$ using nearest-neighbour interpolation.

[1]

Question 4

Paper 1Hard14 marks

An environmental agency models four towns as sites in a Voronoi diagram. Coordinates are in kilometres. Each point in a cell is assigned the annual rainfall measured at that cell’s site (nearest-neighbour interpolation).

Site	Coordinates	Annual rainfall (mm)
A	(0, 0)	820
B	(10, 0)	960
C	(0, 8)	1240
D	(10, 10)	1080

A new monitoring station, E, is later opened at (5, 8). Its annual rainfall is 1500 mm. E is also treated as a town site when screening possible toxic-waste facility locations.

(a)

A point P has coordinates (2, 3). Determine the site whose cell contains P and the rainfall assigned to P.

[2]

(b)

A point Q has coordinates (4, 6). Determine the change in its rainfall estimate when E is added to the Voronoi diagram.

[2]

(c)

After E is added, two relevant Voronoi edge equations are y = 4 and 10x + 16y = 89. Find their point of intersection, W, and use the coordinates of all five sites to determine whether W is a vertex of the updated Voronoi diagram.

[4]

(d)

Before E was added, V₀ = (5.8, 5) was a vertex where the cells of B, C and D met. Determine whether V₀ remains a Voronoi vertex after E is added.

For the agency’s toxic-waste screening model, the search is restricted to the following three triple-edge vertices in its approved search zone. The agency selects the point whose distance from its nearest town is greatest.

Candidate	Coordinates
U	(5, 39/16)
W	(5/2, 4)
X	(91/10, 5)

Determine which candidate the agency should select. Give its distance from the nearest town.

[6]

Question 5

Paper 1Hard15 marks

A council is planning a radio relay, $H$ for three flood-monitoring stations, $A$ $B$ and $C$ The relay is to be placed so that it is the same distance from each station. Figure 1 is a coordinate map; coordinates are in kilometres, with east as the positive $x$ -direction and north as the positive $y$ -direction.

The equation of the line segment $AB$ is $y=\frac13x+\frac53$ and its midpoint is $M(4,3)$

(a)

Write down the coordinates of stations $A$ and $C$ from Figure 1.

[2]

(b)

Show that the equation of the perpendicular bisector of $AB$ is $y=-3x+15$

[3]

(c)

Use your answer to part (b) and the information in Figure 1 to find the coordinates of a point that is the same distance from all three stations. Show enough working to justify that it has this property.

[7]

(d)

The relay can serve the stations only if it is no farther than $3.6\text{ km}$ from each station. Determine whether the point you found in part (c) meets this requirement. Give the distance to three significant figures.

[3]

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