A boat travels from port $A$ to port $B$, a distance of $8$ km on a bearing of $065^\circ$. It then travels from $B$ to port $C$, a distance of $11$ km on a bearing of $145^\circ$.

In triangle $ABC$, $AB=9$ cm, $AC=14$ cm and the area of the triangle is $50\text{ cm}^2$. Angle $BAC$ is obtuse.

Two points $P$ and $Q$ lie on horizontal ground in a straight line with the base $B$ of a vertical tower. Point $Q$ is $35$ m closer to the tower than point $P$. The angles of elevation of the top $T$ of the tower from $P$ and $Q$ are $28^\circ$ and $42^\circ$ respectively.

Two observation points $A$ and $B$ lie on horizontal ground on the same straight line with the foot $T$ of a vertical tower. Point $B$ is between $A$ and $T$, and $AB=20\text{ m}$. The angle of elevation of the top of the tower from $A$ is $30^\circ$, and from $B$ is $45^\circ$.

A sector of a circle has radius $r\text{ cm}$ and angle $\theta$ radians, where $0<\theta<2\pi$. The perimeter of the sector is $12\text{ cm}$ and its area is $8\text{ cm}^2$. The diagram is a non-scale schematic and does not imply that $\theta$ is a minor angle; both minor and reflex sectors are permitted.

Points $A$ and $B$ lie on an east-west coastline, with $B$ due east of $A$ and $AB=12\text{ km}$. A lighthouse $C$ is observed on a bearing of $060^\circ$ from $A$ and on a bearing of $330^\circ$ from $B$.

In a circle with centre $O$, chord $AB$ has length $6\text{ cm}$ and subtends an angle $\angle AOB=\dfrac{2\pi}{3}$.

In three-dimensional space, points $A$, $B$, $C$ and $D$ have coordinates
$A(0,0,0)$, $B(6,0,0)$, $C(0,8,0)$ and $D(0,0,12)$.

A right square pyramid has a horizontal square base of side length $10$ cm. Its apex is vertically above the centre of the base. Each slant edge makes an angle of $58^\circ$ with the base plane.

Two points $A$ and $B$ are on level ground, with $B$ due east of $A$ and $AB=200$ m. The base $D$ of a vertical mast is on a bearing of $035^\circ$ from $A$ and on a bearing of $320^\circ$ from $B$. The angle of elevation of the top $T$ of the mast from $A$ is $22^\circ$.

A right circular cone has slant height $15$ cm and curved surface area $250\text{ cm}^2$. The curved surface can be opened out to form a sector of a circle.

The points $A(2,-1,3)$, $B(8,5,7)$ and $C(5,9,-1)$ are in 3-dimensional space. Let $M$ be the midpoint of $AB$.

A right square-based pyramid has base $ABCD$ of side length $8\text{ cm}$. Its vertex $V$ is vertically above the centre $O$ of the base. The perpendicular height $VO$ is $6\text{ cm}$. Let $M$ be the midpoint of $AB$.

In triangle $ABC$, $AB=8\text{ cm}$, $AC=7\text{ cm}$ and $\angle BAC=120^\circ$. Point $D$ lies inside the triangle such that $AD$ bisects $\angle BAC$ and meets $BC$ at $D$.

A surveying point $B$ is $12\text{ km}$ from a point $A$ on a bearing of $030^\circ$. A point $C$ is due east of $A$ and is on a bearing of $150^\circ$ from $B$. A vertical mast has its base at $C$ and its top at $T$.

A sector $AOB$ has centre $O$, radius $r\text{ cm}$ and angle $\theta$ radians. The arc length $AB$ is $3\pi\text{ cm}$ and the perimeter of the sector is $18+3\pi\text{ cm}$.

In three-dimensional space, points $A$, $B$, $C$ and $D$ have coordinates $A(0,0,0)$, $B(4,0,0)$, $C(0,6,0)$ and $D(0,0,8)$.

A right circular cone has base radius $6\text{ cm}$ and perpendicular height $8\text{ cm}$. The cone is cut by a plane parallel to its base, halfway up its height. The smaller cone above the cut is removed, leaving a frustum with both circular ends exposed.

A glass roof is in the shape of a right rectangular pyramid. Its rectangular base is $18\text{ m}$ by $12\text{ m}$, and its vertex $V$ is vertically above the centre $O$ of the base. The height $VO$ is $7\text{ m}$.

Two observation posts $A$ and $B$ are on level ground, with $B$ due east of $A$ and $AB=1.20\text{ km}$. A weather balloon is vertically above point $C$ on the ground. The bearing of $C$ from $A$ is $035^\circ$ and the bearing of $C$ from $B$ is $300^\circ$. The angle of elevation of the balloon from $A$ is $12^\circ$.

A sculpture is made from two identical right circular cones joined exactly at their circular bases. The common base radius is $6\text{ cm}$ and the distance between the two vertices is $20\text{ cm}$. The sculpture fits exactly inside a sphere centred at the centre of the common base, with both vertices lying on the sphere.

A vertical mast has its base at $O(0,0,0)$ on horizontal ground and its top at $T(0,0,h)$. Three guy wires join $T$ to ground anchors $A(20,0,0)$, $B(-10,18,0)$ and $C(-12,-16,0)$, where distances are measured in metres. The wire $TA$ has length $25\text{ m}$. The diagram is schematic and not drawn to scale; use the coordinates given rather than the apparent positions of the points.

A sector of a circle has radius $r$ cm and angle $\theta$ radians. The perimeter of the sector is $30$ cm and the area of the sector is $53\text{ cm}^2$.

A solid is made from a hemisphere of radius $r$ cm by removing a right circular conical cavity. The base of the cone is the flat circular face of the hemisphere and the apex of the cone is the highest point of the hemisphere.

A roof is a right rectangular pyramid on a horizontal rectangular base measuring $16\text{ m}$ by $10\text{ m}$. The apex is vertically above the centre of the base. The sloping line from the apex to the midpoint of one of the $16\text{ m}$ sides makes an angle of $35^\circ$ with the base plane.

A solid is made by joining two right circular cones base-to-base. The common circular base has radius $5\text{ cm}$. The perpendicular heights of the two cones are $12\text{ cm}$ and $5\text{ cm}$ respectively. The circular base where the cones meet is not exposed.

A sector of a circle has radius $r\text{ cm}$ and angle $\theta$ radians. Its perimeter is fixed at $18\text{ cm}$, where $0<\theta<2\pi$.

Two observation points $A$ and $B$ are on level ground, with $B$ due east of $A$ and $AB=10\text{ km}$. A radio transmitter has base $P$ on the ground. The bearing of $P$ from $A$ is $030^\circ$, and the bearing of $P$ from $B$ is $300^\circ$. The top of the transmitter is $T$, vertically above $P$.

A decorative garden bed is in the shape of a minor circular segment. Its boundary consists of a chord of length $16\text{ m}$ and a minor arc of length $18\text{ m}$. The arc belongs to a circle of radius $r\text{ m}$ and subtends an angle $\theta$ radians at the centre, where $0<\theta<\pi$.

A park has the shape of a quadrilateral $ABCD$. The lengths are $AB=80\text{ m}$, $BC=55\text{ m}$, $CD=70\text{ m}$ and $AD=65\text{ m}$. The angle $BAD$ is $70^\circ$.

A sector of a circle has radius $r\text{ cm}$, angle $\theta$ radians and fixed perimeter $40\text{ cm}$. Assume that $0<\theta\le 2\pi$.