DP IB Maths: AA HL

Topic Questions

3.1 Geometry Toolkit

1a
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2 marks

A competitor is flying their kite in a competition. The kite is on a string of length 206 m and has an angle of elevation of 74 degree from the competitor as shown in the diagram below.

q1a-3-1-hard-ib-ai-sl-maths

Calculate the vertical height, in metres, that the kite is flying at above the point the competitor is holding it.

1b
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3 marks

A second competitor raises their kite to the same vertical height from the same position as the first competitor. The angle between the two kites is 13 degree as shown in the diagram below.

q1b-3-1-hard-ib-ai-sl-maths

Calculate the length of the string for the kite flown by the second competitor.

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2a
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3 marks

A small airline operates between three locations A, B and C, in one particular country. B is located 530 km from A on a bearing of 248 degree. C is located 300 km due East from the midpoint, M, of AB. This information is shown in the diagram below.

q2a-3-1-hard-ib-ai-sl-maths

Calculate the distance AC.

2b
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4 marks

Calculate the bearing that an aeroplane would need to fly on if it were travelling from C to B.

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3a
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6 marks

An artist has been commissioned to create a sculpture for the Mathematics department of a University. She decides to approximate a Fibonacci spiral from a 5 m length of copper wire by putting together a series of squares of increasing size with an arc of a quarter circle in each square. The centre of each arc is at a vertex of the square and the radius is the same as the side length of the square. The copper wire is to be used for the spiral and the edges of the outer rectangle only as shown in the diagrams below.

q3a-3-1-hard-ib-ai-sl-maths

Calculate the length and width of the rectangle that encases the spiral.

3b
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4 marks

In order to make the spiral stand out, various sections of the sculpture are to be filled in with coloured glass as shown below.

q3b-3-1-hard-ib-ai-sl-maths

Calculate the area of glass that is required to complete the sculpture.

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4a
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2 marks

The points A, B and C, located on the grid below, form a triangle.

q4a-4-1-hard-aa-al-maths

Find the length of BC.

4b
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2 marks

Find the coordinates of the midpoint, M, of [BC].

4c
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3 marks

Calculate the area of the triangle ABM.

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5a
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2 marks

A person requires rescuing from the top of a building at a height of 8.2 m. A fire truck has an extendable ladder with its fixed end at a height of 1.6 m. It has been parked at a horizontal distance of 3.7 m from the building, as shown in the diagram below.

q5a-3-1-hard-ib-ai-sl-maths

Calculate the length of the ladder required to reach the top of the building.

5b
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2 marks

For safety purposes, the angle made between the ladder and the horizontal surface it stands on should be between 70 degree minus 80 degree.

Show that the ladder on the fire truck, in this situation, would not be safe.

5c
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2 marks

The fire truck is moved to a horizontal distance from the building that enables the optimal angle of 75 degree to be achieved.

Calculate the length that the ladder now has to be extended to.

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6a
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3 marks

A triangular piece of land has been marked out by placing string around 3 stakes at positions A, B and C as shown in the diagram below. The length AC is 22 m, BC is 14 m and straight A straight B with hat on top straight C is a right angle.

q6a-3-1-hard-ib-ai-sl-maths

Calculate the total length of the string used.

6b
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2 marks

Calculate the area of the piece of land.

6c
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4 marks

The section of land is to be adjusted. Point A and C remain fixed in position but point B is moved until angle straight A straight C with hat on top straight B becomes 90 degree.The overall length of the string does not change.

Calculate the new length of BC.

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7a
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5 marks

The shape ABCDEFG, as seen in the diagram below, shows the footprint of a new building that is to be constructed. ED and FG are parallel, as are CD, AG and EF. BC = 28 m, AB = 20 m, AG = 55 m, EF = 15 m and the perpendicular height of FG is 18 m. Angle straight B straight A with hat on top straight G is 90 degree, angle straight A straight B with hat on top straight C is 65 degree and angle straight E straight F with hat on top straight G is 58 degree.

q7a-3-1-hard-ib-ai-sl-maths

Calculate the area of the footprint of the building.

7b
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4 marks

An internal wall is to be constructed along the line DG.

Work out the length of the internal wall and the angle that it makes with FG.

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8a
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2 marks

A gymnast is competing in the women’s uneven bars event. The bars are held in place by vertical supports at points A and B, as shown in the diagram, where A and B are situated at heights of 2.5 m and 1.7 m above the ground respectively. The horizontal distance between the bars is 1.1 m. This information is shown in the diagram below.

It can be assumed that the gymnast travels in a straight line when moving between points A and B.

q8a-3-1-hard-ib-ai-sl-maths

Calculate the distance the gymnast travels in moving between points A and B.

8b
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2 marks

Calculate the angle of depression from point A to point B.

8c
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4 marks

When the gymnast is hanging vertically from the higher bar with her arms fully extended, there is a distance of 0.6 m between point A and her eye level.

Calculate the difference between the angle of depression calculated in part (b) and the angle of depression that the gymnast sees to point B.

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9
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5 marks

A pizza is divided into slices with an angle of 30 degree. Each slice has an area of fraction numerator 3 pi over denominator 16 end fraction cm2.

Find the perimeter of each slice. Leave your answer in terms of pi.

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10
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5 marks

A new lamp has been designed that comprises an annulus containing the light bulbs on top of a stand in the shape of an equilateral triangle of length 9.6 cm. The supporting edges of the stand are divided into thirds by the inner and outer edges of the light disc connecting to it at equally spaced points. The top vertex of the triangular base is located at the centre of the of the two circles that define the annulus.

[An annulus is a ring shaped object made up of a circle with a concentric circle removed from the centre].

A diagram of the lamp is shown below.

q10a-3-1-hard-ib-ai-sl-maths

Calculate the area of the section of the annulus that can be seen from the front.

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11a
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3 marks

The points straight A comma space straight B comma space straight C spaceand straight D have position vectors:

bold italic a equals 3 bold italic i plus 2 bold italic j plus 7 bold italic k comma space space
bold italic b equals bold italic i plus 5 bold italic j plus bold italic k comma space space
bold italic c equals 4 bold italic i minus 4 bold italic j plus 10 bold italic k space and space space
bold italic d equals 2 bold italic j minus 2 bold italic k.

(i)
Find CD with rightwards arrow on top.
(ii)
Show that AB with rightwards arrow on top equals lambda CD with rightwards arrow on top and state the value of lambda.
11b
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4 marks

A fifth point straight E left parenthesis p comma q comma r right parenthesis  is located such that A,B and E lie on a straight line and vertical line BE vertical line equals 21 units.

Find the value of p comma space q and r.

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12a
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3 marks

Points straight A comma space straight B and straight C have position vectors OA with rightwards arrow on top equals open parentheses table row cell negative 2 end cell row 5 end table close parentheses comma space OB with rightwards arrow on top space equals space open parentheses table row 1 row 3 end table close parentheses space and space OC with rightwards arrow on top equals open parentheses table row 4 row 14 end table close parentheses and  respectively.

Given that ABC is a right-angled triangle, find the area of ABC.

12b
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3 marks

A fourth point, straight D, is to be located such that BD with rightwards arrow on top equals 5 CA with rightwards arrow on top.

Find the coordinates of point straight D .

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13
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6 marks

In the diagram below left square bracket OXA right square bracket comma space left square bracket OYB right square bracket spaceand left square bracket ABC right square bracket are straight lines.

The position vectors of A and B relative to O are 6 bold a and 4 bold b respectively.

Y is the midpoint of left square bracket OB right square bracket and AB with rightwards arrow on top equals 2 BC with rightwards arrow on top.

1 over k space OA with rightwards arrow on top equals XA with rightwards arrow on top.

q12-3-1-hard-ib-aa-hl-maths 

Given that open square brackets XYC close square brackets is a straight line, find the value of k.

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1a
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2 marks

ABCD is an isosceles trapezoid where AB equals 17 space straight m and AD equals BC equals 25 space straight m, as shown in the diagram below.

q1a-3-1-medium-ib-ai-sl-maths

Find the height, h, of the trapezoid.

1b
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4 marks

Find the area of the trapezoid.

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2
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3 marks

The distance between Ho Chi Minh City and Hong Kong is known to be 1500 km. The bearing of Hong Kong from Ho Chi Minh City is 046°. Another city, Brisbane, is 6500 km from Ho Chi Minh City on a bearing of 136°. Calculate the distance between Hong Kong and Brisbane.

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3a
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2 marks

Point straight A has coordinates left parenthesis 4 comma negative 6 right parenthesis and point straight B has coordinates space left parenthesis 8 comma 6 right parenthesis.

Calculate the distance of the line segment AB.

3b
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2 marks

Find the equation of the line connecting points straight A and straight B.

Give your answer in the form y equals m x plus c.

3c
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4 marks
(i)
Find the midpoint of open square brackets AB close square brackets.

(ii)
Find the equation of the perpendicular bisector to the line segment AB.
Give your answer in the form y equals m x plus c.

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4a
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3 marks

The diagram below shows a circle with a 68 degree sector cut from it. The radius of the circle is 5 space cm.

q4a-3-1-medium-ib-ai-sl-maths

Find the length of

(i)
the minor arc AB

(ii)
the major arc AB.
4b
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3 marks

Find the area of the shaded region.

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5a
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4 marks

A lawn sprinkler sprays water over a lawn covering an arc of 160 degree with a maximum spray distance of x m as shown in the diagram below. The lawn sprinkler waters 20 space straight m squared of the lawn.

q5a-3-1-medium-ib-ai-sl-maths

Calculate the value of x.

5b
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3 marks

Calculate the length of the outer arc.

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6a
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4 marks

A windscreen wiper blade is 0.8 space straight m spacelong. When in motion the blade moves through an arc of theta degree and wipes an area of 4 over 15 pi straight m squared.

q6a-3-1-medium-ib-ai-sl-maths

Calculate the value of theta.

6b
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3 marks

Calculate the length travelled by the outer edge of the blade.

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7a
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3 marks

The diagram below shows a dirt racetrack where the straights are 426 space straight m long and the longest distance from one end of the track to the other is 554 straight m.

q7a-3-1-medium-ib-ai-sl-maths

Find the total distance around the racetrack.

7b
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4 marks

Find the total area enclosed by the racetrack.

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8a
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2 marks

The diagram below shows a cookie cutter in the shape of a heart constructed from a triangle and two identical semi circles. The height of the triangle is 8 and its base AB is 13.34 space cm.

q8a-3-1-medium-ib-ai-sl-maths

Find the length of the line AC.

8b
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4 marks

Calculate the total area of the heart.

8c
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2 marks

Bob makes some cookie dough and rolls it out on his kitchen bench. The cookie dough covers 1314 space cm squared.

Find the number of full cookies Bob can cut from the dough.

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9a
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3 marks

The diagram below shows a slice of pizza that forms a sector of a circle with an arc of 60 degree and radius of 15 space cm. The width of the crust is 2.2 space cm.

q9a-3-1-medium-ib-ai-sl-maths

Find the perimeter of the slice of pizza.

9b
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3 marks

Find the area of the crust.

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10a
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4 marks

A parallelogram ABCD is shown in the diagram below.

q10a-3-1-medium-db-aa-hl-maths

AB with rightwards arrow on top equals bold a and AD with rightwards arrow on top space equals space bold b.

A new line is added to the diagram connecting B to D.

A point X lies 2 over 3 of the way along BD with rightwards arrow on top.

Express CX with rightwards arrow on top in terms of a and b.

10b
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3 marks

A new point Y lies on the line CD such that AXY is a straight line.

Express AY with rightwards arrow on top in terms of a and b.

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11a
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4 marks

Three points are located at straight A left parenthesis 0 comma 5 right parenthesis comma space straight B left parenthesis 6 comma 4 right parenthesis and straight C left parenthesis 16 comma 8 right parenthesis .

(i)
Find the magnitude of vector AB with rightwards arrow on top .
(ii)
Find the magnitude of vector BC with rightwards arrow on top .
11b
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2 marks

Given that the angle straight A straight B with hat on top straight C is a right angle, find the area of triangle ABC.

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12a
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5 marks

The points straight A comma space straight B comma space straight C and straight D have position vectors bold a bold comma bold b bold comma bold c and bold d, relative to the origin O.

The position vectors are given by

bold italic a equals 2 i plus 4 j minus k
bold italic b equals negative r i plus j plus 2 k space
bold italic c equals 3 i plus s j space
bold italic d equals 2 i minus 2 j minus t k

where r comma space s and t are constants.

Given that BA with rightwards arrow on top space equals space CD with rightwards arrow on top, find r comma space s and t.

12b
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5 marks

A fifth point, straight E, has position vector bold e, relative to the origin O.

Given that AE with rightwards arrow on top equals 3 stack CD comma with rightwards arrow on top find the position vector of straight E.

12c
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2 marks

Find the unit vector that has the same direction as bold e.

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1a
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2 marks

A security lamp is situated at a height of 2.5 m and positioned so that the central axis of the light bulb is directed perpendicularly to the horizontal. When the lamp is switched on the light spreads out in all directions up to an angle of 38° from the central axis of the light bulb. This information is shown in the diagram below.

q1a-3-1-very-hard-ib-ai-sl-maths

Calculate the horizontal distance on the floor that is illuminated by the lamp.

1b
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4 marks

The area illuminated is not sufficient so the lamp is repositioned at the same height so that the central axis of the light bulb is now at an angle of 70 degree from the horizontal.

q1b-3-1-very-hard-ib-ai-sl-maths

Calculate the percentage increase in the horizontal distance that is now illuminated.

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2a
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4 marks

On the diagram below, points straight A open parentheses negative 2 comma 11 over 2 close parentheses and straight B open parentheses 1 half comma 1 half close parentheses indicate the position of metal spikes called pitons that aid mountaineers as they climb. Each unit on the graph is 16 cm in real life.

q2a-3-1-very-hard-ib-aa-sl-maths

A third piton is required halfway between points A and B at point M.

Calculate the distance AM.

2b
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2 marks

Given that all pitons should be spaced at the distance calculated in part (a), work out the number of pitons required on a mountain with a slope length of 25 space square root of 5 straight m.

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3a
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4 marks

An equilateral triangular jigsaw piece has an edge length of 32 mm. Several of these pieces are connected together with the vertices of the triangular pieces alternately pointing up and then down. The completed jigsaw puzzle is in the shape of a parallelogram with a side length of 64 cm and a perpendicular height of 24 space square root of 3 cm.  A diagram illustrating this information can be seen below.

q3a-3-1-very-hard-ib-ai-sl-maths

Calculate the number of individual jigsaw pieces in the puzzle.

3b
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5 marks

A second jigsaw is to be designed using 289 of the same type of individual pieces.  The completed puzzle will this time be in the shape of an equilateral triangle.

Work out the number of pieces required along each length.

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4a
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2 marks

A roof with a symmetrical triangular cross-section, ABC, is being designed for the top of a building. The horizontal width that the roof must span is 28 m and the lengths of the timbers used for the angled part of the cross-section are 21 m, as shown in the diagram below. 

q4a-3-1-very-hard-ib-ai-sl-maths

Calculate the angle straight C straight A with hat on top straight B.

4b
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2 marks

An alternative design idea for the roof is to shorten AC and to make the apex of the roof a right angle. BC remains the same length as it was originally. These changes can be seen in the diagram below. The point X is situated such that it is directly beneath point C.

q4b-3-1-very-hard-ib-ai-sl-maths

Calculate the new length of AC.

4c
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3 marks

Calculate the vertical height CX of this alternative design for the roof.

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5a
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3 marks

A bird is perched on the edge of a building with its eye at a height of 9.5 m above ground level. A person is holding a sandwich at a height of 1.2 m from the ground and the distance between the ground and the person’s eye level is 1.6 m. A diagram showing this is below.

q5a-3-1-very-hard-ib-ai-sl-maths

The bird sees the sandwich at an angle of depression of 52 degree.

Calculate the distance that the bird must fly to reach the food.

5b
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4 marks

The person’s eyes are 0.3 m further away from the building than the sandwich.

Find the angle of elevation at which the person sees the bird.

5c
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3 marks

A second bird is perched on a lamp post on the other side of the person at a horizontal distance of 5 m. The person sees this bird at an angle of elevation of 37 degree.

Find the vertical distance between the two birds.

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6a
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5 marks

Three locations in a forest, straight A space left parenthesis 3 comma 5 right parenthesis comma space straight B space left parenthesis negative 3 comma 2 right parenthesis and straight C space left parenthesis 6 comma negative 4 right parenthesis space are marked out for an orienteering activity. These can be seen on the grid below. Each unit on the grid indicates a distance of 1 km.

q6a-3-1-very-hard-ib-ai-sl-maths

For AC, find:

(i)
The length AC

(ii)
The bearing of A from C.
6b
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4 marks

Point X is the midpoint between B and the point D. X is located on the line AC such that  AX : XC = 1 : 3.   

Find the coordinates of the point D.

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7a
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2 marks

A wheelchair ramp is required to provide access to a building with a door that is located 22 cm above ground level. The maximum angle that a ramp must be from the horizontal is 4.8 degree.

Calculate the minimum horizontal distance that the ramp must extend out.

7b
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6 marks

The wheelchair ramp is built using the minimum distance found in part (a), rounded to 3 significant figures. The ramp is supported by a steel frame, a cross section of which can be seen in the diagram below. A metal strut joins M, the midpoint of AC, to a point X on the line AB. XM is 11.1 cm in length and straight M straight X with hat on top straight C forms a right angle.

q7a-3-1-very-hard-ib-ai-sl-maths

Calculate the length XB.

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8a
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6 marks

In the diagram below, ABCD shows a piece of geometric art on canvas measuring 58 cm by 78 cm. N is the midpoint of BC and M is the midpoint of AB. X is a point on AC such that  AX : XC = 1 : 5.  A straight line connects M to point X. Y is the point where AC intersects ND.

q8a-3-1-very-hard-ib-ai-sl-maths

Calculate the area of the artwork that is painted black.

8b
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4 marks

The piece of artwork is to be enlarged by a length scale factor of 6 and painted on an exterior wall of an art gallery. A 200 ml tin of paint costs $8 and covers an area of 2.4 m2.

Calculate the cost of the paint that must be purchased to re-create the same black sections from part (a) on the wall.

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9a
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4 marks

The logo of a new company comprises a circle of radius r spacecm and centre straight O, with part of the interior area shaded. The diameter of the unshaded interior semi-circle (the unshaded area below the dashed line in the diagram) is  2/3  that of the larger circle. The remainder of the unshaded area is a sector of the main circle with a sector angle of 124°. This information is shown in the diagram below.

q9a-3-1-very-hard-ib-ai-sl-maths

Show that the area of the shaded section is equal to 13 over 30 pi r squared space space cm squared.

9b
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3 marks

The sector angle of the unshaded sector is decreased.

Find the sector angle of the unshaded sector that is required to make the areas of the logo that are shaded and unshaded equal.

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10a
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3 marks

In the diagram below OA with rightwards arrow on top equals bold a and OB with rightwards arrow on top equals bold b. Y is the midpoint of left square bracket OA right square bracket. C is a point on left square bracket OB right square bracket such that OC with rightwards arrow on top space colon space CB with rightwards arrow on top equals 3 colon 2. .

q10a-3-1-very-hard-3-1-ib-aa-hl-maths

Find

(i)
OC with rightwards arrow on top in terms of bold a and bold b.

 

(ii)
BY with rightwards arrow on top in terms of bold a and bold b.
10b
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4 marks

X is a point on left square bracket AC right square bracket. It is given that BX with rightwards arrow on top equals alpha BY with rightwards arrow on top and CX with rightwards arrow on top equals beta CA with rightwards arrow on top, where alpha comma space beta element of straight real numbers.

Find an expression for  OX with rightwards arrow on top

(i)     in terms of bold a bold comma bold space bold b and alpha.

(ii)   in terms of bold a bold comma bold space bold b and beta.

10c
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4 marks

Hence find the values of alpha and beta.

10d
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3 marks

Write down an expression for XY with rightwards arrow on top  in terms of bold a and bold b only.

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11a
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2 marks

A parallelogram is formed by the points straight A left parenthesis negative 1 comma 7 right parenthesis comma space straight B left parenthesis 2 comma 12 right parenthesis comma straight C and straight D, where AC with rightwards arrow on top equals open parentheses table row 12 row 7 end table close parentheses.

Find the coordinates of C and D.

11b
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4 marks

Calculate the perpendicular height of the parallelogram.

11c
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2 marks

Hence find the area of the parallelogram ABCD.

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12a
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2 marks

Points A and B have coordinates left parenthesis 4 comma negative 7 comma n right parenthesis and left parenthesis negative 1 comma space 3 p comma space 2 right parenthesis respectively.

Given that AB with rightwards arrow on top equals open parentheses table row cell negative 5 end cell row 19 row 6 end table close parentheses , find the value of n and p.

12b
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7 marks

Consider a unit vector bold italic v, such that bold italic v equals 4 over 9 bold space bold italic i plus q bold italic j minus 7 over 9 space bold italic k, where q greater than 0.

Point C is such that AC with rightwards arrow on top space equals space 18 bold italic v.

Find the coordinates of point C.

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