Distance, Midpoint, Volume, Surface Area
Right-Angle Trig, Sine Rule, Cosine Rule, Area of a triangle
Angles of Elevation/Depression
Arc length, Area of a sector
Perpendicular bisectors/Voronoi Diagrams
100
Find the midpoint between these two points: (5,2,1) and (6,4,7)
x-value of midpoint: (5+6)/2 = 5.5
y-value of midpoint: (2+4)/2 = 3
z-value of midpoint: (1+7)/2 = 4
midpoint: (5.5, 3 , 4)
100
Given a right triangle where one of the other angles measures 50° and the side length opposite that angle measures 10 cm, find the length of the hypotenuse.
H = 10/sin50
H = 13.1 cm
100
Two fixed points, A and B, are 40 m apart on horizontal ground. Two straight ropes, AP and BP, are attached to the same point, P, on the base of a hot air balloon which is vertically above the line AB. The length of BP is 30 m and angle BAP is 48°. Copy the diagram down and draw and label with an x the angle of depression of B from P.
100
Find the area of a sector with a central angle of 60° and a diameter of 10 m.
radius = 10/2 = 5
area = (60/360)*𝝅*52 = 13.1 m2
100
The line of a perpendicular bisector has a gradient (slope) that is the _______________ _______________ of the the line it is perpendicular to.
opposite reciprocal
200
A manufacturer makes trash cans in the form of a cylinder with a hemispherical top. The trash can has a height of 70 cm. The base radius of both the cylinder and the hemispherical top is 20 cm. Find the total volume of the trash can.
Volume of hemisphere (1/2 volume of sphere) + volume of cylinder
1/2*(4/3*𝝅*203) + 𝝅*202*50 = 79600 cm3
200
The diagram shows points in a park viewed from above, at a specific moment in time.
The distance between two trees, at points A and B, is 6.36 m.
Odette is playing football in the park and is standing at point O, such that AÔB= 10° , OA=25.9 m and OÂB is obtuse.
Calculate the size of angle ABO.
Use the sine rule
sin10/6.36 = sinB/25.9
use inverse sine to solve for angle B
45.0°
200
The owner of a convenience store installs two security cameras, represented by points C1 and C2. Both cameras point towards the centre of the store’s cash register, represented by the point R.
The following diagram shows this information on a cross-section of the store.
The cameras are positioned at a height of 3.1 m, and the horizontal distance between the cameras is 6.4 m. The cash register is sitting on a counter so that its centre, R, is 1.0 m above the floor.
The distance from Camera 1 to the centre of the cash register is 2.8 m.
Determine the angle of depression from Camera 1 to the centre of the cash register. Give your answer in degrees.
sinΘ = 2.1/2.8 or tanΘ = 2.1/1.85...
use inverse trig to solve for Θ
Θ = 48.6°
200
The straight metal arm of a windscreen wiper on a car rotates in a circular motion from a pivot point, O, through an angle of 140°. The windscreen is cleared by a rubber blade of length 46 cm that is attached to the metal arm between points A and B. The total length of the metal arm, OB, is 56 cm.
The part of the windscreen cleared by the rubber blade is shown unshaded in the following diagram.
Calculate the length of the arc made by B, the end of the rubber blade.
(140/360) * 2𝝅 * 56 = 137 cm
200
The boundaries between cells are created by drawing the _________________ ___________________ of the line segments connecting the sites.
perpendicular bisectors
300
A vertical pole stands on horizontal ground. The bottom of the pole is taken as the origin, O, of a coordinate system in which the top, F, of the pole has coordinates (0, 0, 5.8). All units are in metres.
The pole is held in place by ropes attached at F.
One of the ropes is attached to the ground at a point A with coordinates (3.2, 4.5, 0). The rope forms a straight line from A to F. Find the length of the rope connecting A to F.
d = √(0-3.2)2 + (0-4.5)2 + (5.8-0)2 = 8.01m
300
Calculate the area of triangle AOB.
use the alternate area formula:
1/2×25.9×6.36×sin (124.996…°) = 67.5 m2
the angle 124.996 is found using the angle sum theorem
300
Calculate the distance from Camera 2 to the centre of the cash register.
Find the missing side length on the triangle with camera 1 and the register using Pythagorean theorem or right-angle trig. √2.82 - 2.12 = 1.85...
6.4 - 1.85... = 4.55 m
Now you know two sides of the triangle with camera 1 and can use the Pythagorean theorem to find the distance from camera 2 to the register (the hypotenuse)
√4.552 + 2.12 = 5.01 m
300
Joey is making a party hat in the form of a cone. The hat is made from a sector, AOB, of a circular piece of paper with a radius of 18 cm and AÔB = Θ as shown in the diagram.
To make the hat, sides [OA] and [OB] are joined together. The hat has a base radius of 6.5 cm. Write down the perimeter of the base of the hat in terms of 𝝅.
2*𝝅*6.5 = 13𝝅
the perimeter of the base is the circumference of the circle
300
Six restaurant locations (labelled A, B, C, D, E and F) are shown, together with their Voronoi diagram. All distances are measured in kilometres. Elena wants to eat at the closest restaurant to her. Which restaurant she should go to, if she is at (2,7)?
Restaurant B
400
A factory packages coconut water in cone-shaped containers with a base radius of 5.2 cm and a height of 13 cm. The factory designers are currently investigating whether a cone-shaped container can be replaced with a cylinder-shaped container with the same radius and the same total surface area. Find the height, ℎ, of this cylinder-shaped container.
Slant height of cone found using Pythagorean theorem: 14 cm
Surface area of cone: 314 cm
2*𝝅*5.2*h + 2*𝝅*5.22 = 314 and solve for h
h = 4.41 cm
400
The Tower of Pisa is well known worldwide for how it leans.
Giovanni visits the Tower and wants to investigate how much it is leaning. He draws a diagram showing a non-right triangle, ABC.
On Giovanni’s diagram the length of AB is 56 m, the length of BC is 37 m, and angle ACB is 60°. AX is the perpendicular height from A to BC.
Use Giovanni’s diagram to show that angle ABC, the angle at which the Tower is leaning relative to the
horizontal, is 85° to the nearest degree.
First, you must find angle CAB, or the total angle A, by using the sine rule. sinCAB/37 = sin60/56. Use inverse sine to find angle CAB which is 34.9...° Then use the angle sum theorem to find angle ABC, so 180 - 60 - 34.9... = 85.096...° which rounds to 85°
400
Without further calculation, determine which camera has the largest angle of depression to the centre of the cash register. Justify your response.
Camera 1 is closer to the cash register than camera 2 and both cameras are at the same height on the wall thus the larger angle of depression is from camera 1.
400
Determine the area of the windscreen that is cleared by the rubber blade.
Find the area of the total sector with radius 56 and subtract the area of the sector not cleared by the base of the windshield wiper with a radius 10.
[(140/360)*𝝅*562] - [(140/360)*𝝅*102] = 3710 cm2
400
Restaurant C is at (7, 8) and restaurant D is at (7, 5). Find the equation of the boundary between these two sites. 
y - 6.5 = 0(x - 7) or y = 6.5
500
Helen is building a cabin using cylindrical logs of length 2.4 m and radius 8.4 cm. A wedge is cut from one log and the cross-section of this log is illustrated in the following diagram. Find the volume of this log.
Formula for volume of log = Volume of cylinder - volume of empty sector
**notice difference in units, you must convert m to cm, so the height of the cylinder is 240 cm
240*𝝅*8.42 - 240*(50/360)*𝝅*8.42 = 45800 cm3
500
The following diagram shows a park bounded by a fence in the shape of a quadrilateral ABCD. A straight path crosses through the park from B to D.
AB=85 m, AD=85 m, BC=40 m, angle CBD=41°, angle BCD=120°.
Calculate the size of angle BAD, correct to five significant figures.
First find angle BDC using the angle sum theorem (19°). The find length BD using the sine rule (106.401). Now you know three side lengths and can use the cosine rule.
cosA = (852 + 852 - 106.4012)/(2*85*85)
use inverse cos to find A, which is 77.495°
500
A hot-air balloon remains at a constant height as it moves further away from a building.
Describe, in words, the change in the angle of depression from the hot air balloon to the top of the building as the horizontal distance between the balloon and the building increases.
The angle of depression from the hot air balloon gets smaller.
500
Find the value of Θ.
The perimeter of the base is the same as the arc length of the arc in the first diagram.
13𝝅 = (Θ/360) *2𝝅*18 and solve for Θ
Θ = 130°
500
What is the distance from the vertex between Restaurants B,C, and D to Restaurant D. Restaurant B is at (3,6), Restaurant C is at (7,8) and Restaurant D is at (7,5).
The distance between D (7,5) and the intersection of the perpendicular bisectors between CD (y = 6.5) and BC (y = -2x + 17), or (5.25, 6.5)
d = √(7 - 5.25)2 + (5 - 6.5)2 = 2.30 km