Geometry Toolkit (DP IB Maths: AI HL)

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  from the competitor as shown in the diagram below.

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

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  as shown in the diagram below.

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

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 . C is located 300 km due East from the midpoint, M, of AB. This information is shown in the diagram below.

Calculate the distance AC.

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

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.

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

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

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

An environmentalist is mapping the trees that are to be protected in a local forest area. Tree A is located at point (4, 5) on his grid and tree B at point (2,9) as shown in the diagram below.

Find the midpoint, M, of AB.

Calculate the distance between the two trees.

A third tree, C, is located on the -axis of the grid. MC is perpendicular to AB.

Find the y-coordinate of C.

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.

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

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

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

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

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

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  is a right angle.

Calculate the total length of the string used.

Calculate the area of the piece of land.

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

Calculate the new length of BC.

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  is , angle  is  and angle  is .

Calculate the area of the footprint of the building.

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.

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.

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

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

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.

Town A is located at (3, 1) and Town B at (7, 13) and a straight section of river runs between the two towns. This information is represented on the diagram below.

Calculate the gradient of the line that models the river.

A bridge is to be built so that it is perpendicular to the river and located at a point exactly halfway between the two towns.

State the coordinates of the midpoint of the river between the two towns.

State the gradient of the line along which the bridge would lie.

Find the equation of the line along which the bridge would lie. Give your answer in the form , where . 

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.

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

The points and  have position vectors:

Find .

Show that  and state the value of .

A fifth point  is located such that A,B and E lie on a straight line and  units.

Find the value of  and .

Points  and have position vectors  and  respectively.

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

A fourth point, , is to be located such that

Find the coordinates of point .

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.

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

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 from the horizontal.

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

On the diagram below, points  and  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.

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

Calculate the distance AM.

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 .

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 .  A diagram illustrating this information can be seen below.

Calculate the number of individual jigsaw pieces in the puzzle.

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.

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. 

Calculate the angle .

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.

Calculate the new length of AC.

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

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.

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

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

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.

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 .

Find the vertical distance between the two birds.

Three locations in a forest,  and  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.

For AC, find:

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.

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 .

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

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  forms a right angle.

Calculate the length XB.

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.

Calculate the area of the artwork that is painted black.

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.

The logo of a new company comprises a circle of radius cm and centre , 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.

Show that the area of the shaded section is equal to .

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.

In the diagram below  and . Y is the midpoint of . C is a point on  such that .

Find

in terms of  and .

in terms of  and .

X is a point on . It is given that  and , where

Find an expression for  

(i)     in terms of  and .

(ii)   in terms of  and .

Hence find the values of  and .

Write down an expression for  in terms of  and  only.

A parallelogram is formed by the points  and , where .

Find the coordinates of C and D.

Calculate the perpendicular height of the parallelogram.

Hence find the area of the parallelogram .

Points A and B have coordinates  and respectively.

Given that , find the value of  and .

Consider a unit vector , such that , where .

Point C is such that .

Find the coordinates of point C.