Two observation points $A$ and $B$ lie on straight horizontal ground, with $B$ located $40.0\ \text{m}$ farther from a vertical tower than $A$. The angles of elevation to the top of the tower from $A$ and $B$ are $51^\circ$ and $34^\circ$ respectively. The observer's eye level is $1.60\ \text{m}$ above the ground.

A window is shaped as a minor circular segment of a circle with centre $O$ and radius $18.0\ \text{cm}$. The radii to the endpoints $A$ and $B$ of the segment form an angle of $110^\circ$.

A fan-shaped sign is a sector with central angle $135^\circ$. Its perimeter, consisting of two radii and one arc, is $52.0\ \text{cm}$. The radius is $r\ \text{cm}$.

Two people stand at points $P$ and $Q$ on straight horizontal ground, $120\ \text{m}$ apart. A drone is vertically above a point between $P$ and $Q$. The angles of elevation of the drone from $P$ and $Q$ are $32^\circ$ and $47^\circ$ respectively.

Two stations $A$ and $B$ lie on a circular railway track with centre $O$. The straight-line distance $AB$ is $24.0\ \text{km}$ and the minor angle $AOB$ is $100^\circ$.

A quadrilateral $ABCD$ is divided by diagonal $AC$. In triangle $ABC$, $AB=50.0\ \text{m}$, $BC=65.0\ \text{m}$ and $\angle ABC=72^\circ$. Also, $AD=58.0\ \text{m}$ and $CD=74.0\ \text{m}$.

From harbour $P$, an island $A$ is $18.0\ \text{km}$ away on a bearing of $075^\circ$. A patrol boat travels from $P$ to point $Q$, a distance of $26.0\ \text{km}$ on a bearing of $145^\circ$.

Surveyors mark two points $A$ and $B$ on a straight path, where $AB=240\ \text{m}$. An inaccessible rock formation is at $C$. They measure $\angle BAC=52^\circ$ and $\angle ABC=67^\circ$. A protected circular region of radius $60\ \text{m}$ is centred at $C$; within triangle $ABC$, its boundary forms a sector.

A Ferris wheel has radius $18\ \text{m}$ and its centre is $20\ \text{m}$ above horizontal ground. A passenger moves through a central angle of $140^\circ$, starting at the lowest point of the wheel.

The cross-section of a greenhouse is an isosceles triangle $ABC$, where $BC=12.0\ \text{m}$ and $AB=AC=8.00\ \text{m}$. A ventilation opening is a sector centred at $A$, with radius $2.50\ \text{m}$ and central angle $\angle BAC$.

A straight zipline runs from a platform at $A$, which is $24.0\ \text{m}$ above horizontal ground, to a landing point $B$ on the ground. The horizontal distance from the point directly below $A$ to $B$ is $80.0\ \text{m}$. A support point $C$ is on the zipline, horizontally $50.0\ \text{m}$ from the point below $A$.

A rotating floodlight illuminates an annular sector between distances $4.00\ \text{m}$ and $13.0\ \text{m}$ from its pivot. During one programmed movement, it turns through $150^\circ$.

An orienteer travels $14.0\ \text{km}$ from checkpoint $A$ to checkpoint $B$ on a bearing of $035^\circ$. The orienteer then travels $19.0\ \text{km}$ from $B$ to checkpoint $C$ on a bearing of $110^\circ$.

A flower bed is in the shape of a sector with central angle $96^\circ$. Its perimeter, consisting of two radii and the minor arc, is $46.0\ \text{m}$. Let the radius be $r\ \text{m}$.

A quadrilateral wetland $ABCD$ is divided by diagonal $AC$. Measurements give $AB=90.0\ \text{m}$, $BC=70.0\ \text{m}$, $\angle ABC=58^\circ$, $AD=65.0\ \text{m}$ and $\angle DAC=44^\circ$.

A lakeside path follows the major arc between points $A$ and $B$ on a circle of radius $30.0\ \text{m}$. The straight-line distance $AB$ is $42.0\ \text{m}$. Let $\theta$ be the angle subtended by the minor arc at the centre $O$.

Two radio receivers $A$ and $B$ are $12.0\ \text{km}$ apart, with $B$ due east of $A$. A transmitter $P$ is observed from $A$ on a bearing of $028^\circ$ and from $B$ on a bearing of $318^\circ$. A circular restricted zone of radius $2.50\ \text{km}$ is centred at $P$.

Two cameras are located at $A$ and $B$ on one side of a stadium, where $AB=150\ \text{m}$. A performer at $T$ is observed such that $\angle BAT=73^\circ$ and $\angle ABT=41^\circ$. A circular safety zone of radius $25.0\ \text{m}$ is centred at $T$.

A triangular solar array $ABC$ has side lengths $AB=13.0\ \text{m}$, $BC=14.0\ \text{m}$ and $AC=15.0\ \text{m}$. A sector of radius $4.00\ \text{m}$ is removed at the vertex with the largest angle. The diagram is not drawn to scale.

A triangular wildlife reserve $ABC$ is surveyed from a straight boundary $AB$. The measurements are $AB=180\ \text{m}$, $\angle BAC=47^\circ$ and $\angle ABC=68^\circ$. A circular sanctuary of radius $35\ \text{m}$ is centred at $C$. Within the reserve, the sanctuary forms a sector bounded by $CA$ and $CB$.

An aircraft flies $32.0\ \text{km}$ from airport $A$ to point $B$ on a bearing of $035^\circ$. It then flies $24.0\ \text{km}$ from $B$ to point $C$ on a bearing of $122^\circ$.

Two tracking stations $P$ and $Q$ are $250\ \text{m}$ apart on level ground. A balloon is vertically above a point between the stations. The angles of elevation from $P$ and $Q$ are $28^\circ$ and $41^\circ$ respectively. Let the horizontal distance from $P$ to the point below the balloon be $x\ \text{m}$ and let the balloon's height be $h\ \text{m}$.

A mechanical linkage consists of two rigid arms of lengths $12.0\ \text{cm}$ and $17.0\ \text{cm}$ joined at a fixed pivot. The $12.0\ \text{cm}$ arm is held stationary while the $17.0\ \text{cm}$ arm rotates about the pivot. The angle between the arms is $\theta$, and the distance between their free endpoints is $d\ \text{cm}$.

A wind turbine has three blades of length $42.0\ \text{m}$, equally spaced around the hub. The turbine rotates at a constant rate of $18.0$ revolutions per minute.

A tidal basin is modelled as an annular sector with central angle $72^\circ$. At low tide, the water extends $85.0\ \text{m}$ from the gate at the centre. At high tide, it extends $120\ \text{m}$ from the gate.

A cable-car route connects lower station $A$ to upper station $B$. Relative to $A$, station $B$ is $900\ \text{m}$ horizontally away and $240\ \text{m}$ higher. A proposed route passes through station $C$, located $600\ \text{m}$ from $A$ at an angle of elevation of $22^\circ$.

A warehouse robot turns along a circular arc of radius $6.00\ \text{m}$ through an angle of $135^\circ$. The robot's initial and final positions on the turn are $A$ and $B$.

The top of a decorative gate is a minor circular arc $AB$. The chord $AB$ has length $16.0\ \text{m}$ and subtends an angle of $76^\circ$ at the centre $O$.

A research vessel travels $22.0\ \text{km}$ from $O$ to $P$ on a bearing of $020^\circ$. It then travels a distance $x\ \text{km}$ from $P$ on a bearing of $130^\circ$ to $Q$. Point $Q$ is due east of $O$.

Points $A$ and $B$ lie on a straight coastline and are $320\ \text{m}$ apart. An offshore platform $C$ is observed such that $\angle BAC=64^\circ$ and $\angle ABC=49^\circ$. Point $M$ is the midpoint of $AB$.

A snowmobile travels $18.0\ \text{km}$ from base $A$ to refuge $B$ on a bearing of $070^\circ$. It then travels $24.0\ \text{km}$ from $B$ to checkpoint $C$ on a bearing of $195^\circ$. A circular shelter zone of radius $3.00\ \text{km}$ is centred at $B$.

An amphitheatre has concentric rows of seats, each forming an arc with central angle $110^\circ$. The first row has radius $8.00\ \text{m}$, and each subsequent radius is $0.750\ \text{m}$ greater. Seats are $0.550\ \text{m}$ wide. Only complete seats can be installed in each row.