Geometry and Trigonometry I

Geometry marks are won by naming the rule you used. "Angles on a straight line add to 180°" earns credit that a bare number does not, so write the reason every time.

Angles and parallel lines

The basic rules:

• Angles on a straight line add to $180°$.
• Angles around a point add to $360°$.
• Vertically opposite angles are equal.
• Angles in a right angle add to $90°$ (complementary); angles adding to $180°$ are supplementary.

Two angles lie on a straight line and one is $125°$. Find the other.

$$180° - 125° = 55°$$

When a line crosses two parallels

A line cutting two parallel lines is a transversal, and it creates three named relationships:

So if one angle is $110°$, the corresponding and alternate angles are also $110°$, while the co-interior angle is $180° - 110° = 70°$.

Triangles and polygons

The angles of a triangle add to $180°$.

Two angles of a triangle are $47°$ and $68°$. Find the third.

$$180° - 47° - 68° = 180° - 115° = 65°$$

Triangle types worth naming:

• Equilateral: three equal sides, all angles $60°$.
• Isosceles: two equal sides, and the angles opposite them are equal.
• Scalene: no equal sides or angles.
• Right-angled: one angle of $90°$.

Polygons

For a polygon with $n$ sides:

$$\text{sum of interior angles} = (n-2) \times 180°$$ $$\text{sum of exterior angles} = 360° \text{, always}$$

A pentagon has $n = 5$, so its interior angles add to $(5-2) \times 180° = 540°$.

For a regular polygon every angle is the same, so:

$$\text{each exterior angle} = \frac{360°}{n} \qquad \text{each interior angle} = 180° - \frac{360°}{n}$$

A regular hexagon has exterior angle $\frac{360}{6} = 60°$ and interior angle $180 - 60 = 120°$.

Parts of a circle

Two properties used constantly:

• A tangent is perpendicular to the radius drawn to the point of contact.
• A perpendicular from the centre to a chord bisects that chord.

Pythagoras' theorem

In a right-angled triangle, with $c$ the hypotenuse (the side opposite the right angle):

$$a^2 + b^2 = c^2$$

The two shorter sides are 6 cm and 8 cm. Find the hypotenuse.

$$c^2 = 6^2 + 8^2 = 36 + 64 = 100$$ $$c = 10 \text{ cm}$$

The hypotenuse is 13 cm and one shorter side is 5 cm. Find the other.

Here you are finding a shorter side, so subtract:

$$a^2 = 13^2 - 5^2 = 169 - 25 = 144$$ $$a = 12 \text{ cm}$$

Trigonometric ratios

In a right-angled triangle, label the sides relative to the angle you are working with: the hypotenuse is opposite the right angle, the opposite faces your angle, and the adjacent is the remaining side next to it.

$$\sin\theta = \frac{\text{opp}}{\text{hyp}} \qquad \cos\theta = \frac{\text{adj}}{\text{hyp}} \qquad \tan\theta = \frac{\text{opp}}{\text{adj}}$$

Remember it as SOH CAH TOA.

Finding a side. A right-angled triangle has hypotenuse 10 cm and an angle of $35°$. Find the side opposite that angle.

Opposite and hypotenuse means sine:

$$\sin 35° = \frac{x}{10}$$ $$x = 10 \sin 35° = 5.74 \text{ cm (3 s.f.)}$$

Finding an angle. The opposite side is 6 cm and the adjacent is 8 cm.

Opposite and adjacent means tangent:

$$\tan\theta = \frac{6}{8} = 0.75$$ $$\theta = \tan^{-1}(0.75) = 36.9° \text{ (1 d.p.)}$$

Transformations

Four transformations, each needing specific information to describe it fully.

Translation — a slide, described by a column vector $\begin{pmatrix} a \\ b \end{pmatrix}$: $a$ across, $b$ up.

Reflection — a flip in a mirror line, which must be named:

• in the $x$-axis: $(x, y) \mapsto (x, -y)$
• in the $y$-axis: $(x, y) \mapsto (-x, y)$
• in the line $y = x$: $(x, y) \mapsto (y, x)$

Rotation — needs a centre, an angle and a direction. About the origin:

• $90°$ anticlockwise: $(x, y) \mapsto (-y, x)$
• $90°$ clockwise: $(x, y) \mapsto (y, -x)$
• $180°$: $(x, y) \mapsto (-x, -y)$

Enlargement — needs a centre and a scale factor $k$. About the origin, $(x, y) \mapsto (kx, ky)$.

Under an enlargement of scale factor $k$, lengths multiply by $k$ but areas multiply by $k^2$. Doubling the sides of a shape gives four times the area.