4. Geometry

Congruent shapes have the same shape and size. Similar shapes have the same shape but may differ in size; every corresponding length changes by one scale factor while corresponding angles stay equal.

Name common polygons by side count: pentagon 5, hexagon 6, octagon 8 and decagon 10. A diagonal joins two non-adjacent vertices.

A quadrilateral with two pairs of parallel sides and all four sides equal is a rhombus. If it also has four right angles, the more specific name is square. A solid with a trapezium cross-section unchanged through its length is a trapezoidal prism.

If $AB$ joins two points on the circumference, it is a chord. If it passes through the centre, it is also a diameter. The region swept out between radii $OA$ and $OB$ is a sector, while the region cut off by chord $AB$ is a segment.

For a reflex angle, first measure the smaller angle between the arms, then subtract it from $360^\circ$. For example, if the smaller angle is $137^\circ$, the reflex angle is $360^\circ-137^\circ=223^\circ$.

1. Draw one given side accurately with a ruler.
2. Put the compass point at one endpoint and set its width to the second side length; draw an arc.
3. Without estimating, reset the compass to the third side length, centre it at the other endpoint and draw an intersecting arc.
4. Use a ruler to join the arc intersection to both endpoints.
5. Leave both construction arcs visible.

For sides $5$ cm, $8$ cm and $10$ cm, draw the $10$ cm base. From one endpoint draw an arc of radius $5$ cm and from the other an arc of radius $8$ cm. Their intersection is the third vertex; joining it to the base endpoints fixes all three lengths.

A cuboid measuring $6$ cm by $3$ cm by $2$ cm needs rectangles in three matching pairs: $6\times3$, $6\times2$ and $3\times2$. Its surface area is $2(18+12+6)=72$ cm$^2$, and its volume is $6\times3\times2=36$ cm$^3$.

At a scale of $1:40\,000$, $1$ cm on the map represents $40\,000$ cm $=400$ m $=0.4$ km. A map length of $7.5$ cm therefore represents $7.5\times0.4=3$ km; a real distance of $5$ km would be drawn as $5\div0.4=12.5$ cm.

The cardinal bearings are north $000^\circ$, east $090^\circ$, south $180^\circ$ and west $270^\circ$. A ruler must be used for every straight direction line.

Reverse bearings differ by $180^\circ$. If the bearing of $B$ from $A$ is $025^\circ$, the bearing of $A$ from $B$ is $025^\circ+180^\circ=205^\circ$. If adding $180^\circ$ passes $360^\circ$, subtract $180^\circ$ instead.

Two similar triangles have corresponding sides $4.5$ cm and $9$ cm, so the scale factor from the smaller to the larger is $9\div4.5=2$. A side corresponding to $3.3$ cm therefore has length $3.3\times2=6.6$ cm in the larger triangle.

A small bottle holds $0.4$ L and a similar large bottle holds $1.35$ L. The linear factor from large to small is $\sqrt[3]{0.4/1.35}$. If the large bottle is $29.7$ cm high, the small height is $29.7\sqrt[3]{0.4/1.35}=19.8$ cm.

If $BE\parallel CD$ in triangle $ACD$, then $\angle ABE=\angle ACD$ and $\angle AEB=\angle ADC$ by corresponding angles; $\angle BAE$ is the common angle at $A$. Therefore triangles $ABE$ and $ACD$ are similar by AA, and their matching sides share one scale factor.

A regular decagon matches after every $36^\circ$, so its rotational order is $360\div36=10$; it also has 10 lines of symmetry. A rhombus has its two diagonals as mirror lines and rotational order 2.

A regular square-based pyramid has four vertical planes of symmetry, each passing through the apex and a symmetry line of the square base. Its rotational axis joins the apex to the centre of the base and has order 4.

In isosceles triangle $ABC$, let $AB=AC$ and $\angle BAC=38^\circ$. Then $\angle ABC=\angle BCA=(180^\circ-38^\circ)\div2=71^\circ$. If $BC$ is extended to $D$, $\angle ACD=180^\circ-71^\circ=109^\circ$ because angles on a straight line sum to $180^\circ$.

If parallel lines $AB$ and $CD$ are cut by transversal $EF$ and one acute angle is $38^\circ$, its alternate and corresponding acute angles are also $38^\circ$. Each adjacent obtuse angle is $180^\circ-38^\circ=142^\circ$; the matching co-interior pair is supplementary.

A hexagon has interior sum $(6-2)\times180^\circ=720^\circ$. If five interior angles are each $115^\circ$, the sixth is $720^\circ-5\times115^\circ=145^\circ$. A regular 24-gon has exterior angle $360^\circ\div24=15^\circ$ and interior angle $165^\circ$.

Let $AB$ be a diameter, let $C$ lie on the circle, and let a tangent touch the circle at $A$. If $\angle BAC=32^\circ$, then $\angle ACB=90^\circ$ because it is an angle in a semicircle. Hence $\angle ABC=180^\circ-90^\circ-32^\circ=58^\circ$. The angle between the tangent at $A$ and chord $AC$ is also $58^\circ$ by the alternate segment theorem.

Mark the centre, chord midpoints, perpendicular signs, radii, tangent contact points and the common external point. State the circle property first, then use congruence, Pythagoras or ordinary angle facts only as separate justified steps.

In one circle, equal chords $AB$ and $CD$ have perpendicular distances $OM=5$ cm and $ON$ from centre $O$. Then $ON=5$ cm because equal chords are equidistant from the centre. If tangents from $T$ touch at $P$ and $Q$ and $TP=8$ cm, then $TQ=8$ cm because tangents from the same external point are equal.