Geometry and trigonometry flashcards

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State the key angle facts for lines, points, triangles and quadrilaterals.
Angles on a straight line add to 180°; angles around a point to 360°; vertically opposite angles are equal; angles in a triangle add to 180°; in a quadrilateral to 360°.
Name the angle pairs formed when a line crosses parallel lines.
Alternate angles (Z shape) are equal; corresponding angles (F shape) are equal; co-interior angles (C or U shape) add to 180°.
How do you find the interior angle sum of a polygon, and what is it for a hexagon?
Sum \(=(n-2)\times180^\circ\). Hexagon: \(4\times180=720^\circ\). The exterior angles of any polygon add to \(360^\circ\).
A regular polygon has exterior angle 30°. How many sides does it have, and what is each interior angle?
Sides \(=360\div30=12\). Interior angle \(=180-30=150^\circ\).
State Pythagoras' theorem and use it with legs 8 cm and 15 cm.
In a right-angled triangle \(a^2+b^2=c^2\), with \(c\) the hypotenuse. \(c=\sqrt{64+225}=\sqrt{289}=17\text{ cm}\). To find a shorter side, subtract.
What do sin, cos and tan mean in a right-angled triangle?
\(\sin\theta=\dfrac{\text{opp}}{\text{hyp}}\), \(\cos\theta=\dfrac{\text{adj}}{\text{hyp}}\), \(\tan\theta=\dfrac{\text{opp}}{\text{adj}}\) (SOH CAH TOA). Label the sides relative to the angle first.
A right-angled triangle has opposite side 7 cm and hypotenuse 25 cm. Find the angle.
\(\sin\theta=\dfrac7{25}=0.28\), so \(\theta=\sin^{-1}(0.28)=16.3^\circ\) (1 d.p.). Check the calculator is in degrees.
(Higher) Write down the exact values of sin, cos and tan for 30°, 45° and 60°.
\(\sin30=\tfrac12\), \(\cos30=\tfrac{\sqrt3}2\), \(\tan30=\tfrac1{\sqrt3}\). \(\sin45=\cos45=\tfrac{\sqrt2}2\), \(\tan45=1\). \(\sin60=\tfrac{\sqrt3}2\), \(\cos60=\tfrac12\), \(\tan60=\sqrt3\).
How are bearings measured? What is the back bearing of 065°?
Bearings are measured clockwise from north and written with three figures. The back bearing adds or subtracts 180°: 065° + 180° = 245°.
(Higher) State the sine rule and when to use it. Find b if A = 40°, a = 8 cm, B = 65°.
\(\dfrac a{\sin A}=\dfrac b{\sin B}\); use it with a side and its opposite angle known. \(b=\dfrac{8\sin65^\circ}{\sin40^\circ}=11.3\text{ cm}\).
(Higher) State the cosine rule and find a if b = 7, c = 8 and A = 60°.
\(a^2=b^2+c^2-2bc\cos A\); use it with two sides and the included angle, or three sides. \(a^2=49+64-56=57\), so \(a=7.55\text{ cm}\).
(Higher) How do you find the area of a triangle from two sides and the included angle?
Area \(=\tfrac12ab\sin C\). For \(a=6\), \(b=9\), \(C=50^\circ\): \(\tfrac12\times54\times\sin50^\circ=20.7\text{ cm}^2\).
(Higher) Find the length of the diagonal of a 3 cm by 4 cm by 12 cm cuboid.
Use Pythagoras in three dimensions: \(\sqrt{3^2+4^2+12^2}=\sqrt{169}=13\text{ cm}\). For the angle between a line and a plane, draw the right-angled triangle containing both.
Name the parts of a circle: chord, tangent, arc, sector, segment.
A chord joins two points on the circle; a tangent touches it at one point; an arc is part of the circumference; a sector lies between two radii and an arc; a segment lies between a chord and an arc.
Find the circumference and area of a circle of radius 6 cm, in terms of π.
\(C=2\pi r=12\pi\text{ cm}\); \(A=\pi r^2=36\pi\text{ cm}^2\). Circumference is a length; area is in square units.
Find the arc length and sector area for a sector of angle 80° and radius 9 cm.
Fraction of circle \(=\dfrac{80}{360}=\dfrac29\). Arc \(=\tfrac29\times2\pi\times9=4\pi=12.6\text{ cm}\). Area \(=\tfrac29\times\pi\times81=18\pi=56.5\text{ cm}^2\).
State two circle theorems about a radius and about a semicircle.
A tangent is perpendicular to the radius at the point of contact. The angle in a semicircle (subtended by a diameter) is 90°.
(Higher) State the circle theorems for angles at the centre, in the same segment and in a cyclic quadrilateral.
The angle at the centre is twice the angle at the circumference on the same arc. Angles in the same segment are equal. Opposite angles of a cyclic quadrilateral add to 180°.
(Higher) State the alternate segment theorem.
The angle between a tangent and a chord equals the angle in the alternate segment, which is the segment on the opposite side of the chord from that angle.
What are the four conditions for triangles to be congruent?
SSS (three sides), SAS (two sides and the included angle), ASA or AAS (two angles and a corresponding side), RHS (right angle, hypotenuse and one side).
(Higher) If shapes are similar with scale factor k, how do areas and volumes scale?
Lengths scale by \(k\), areas by \(k^2\), volumes by \(k^3\). Heights 4 cm and 6 cm, smaller volume 80 cm³: \(k=1.5\), larger volume \(=80\times1.5^3=270\text{ cm}^3\).
Give the area formulae for a triangle, parallelogram and trapezium, and find a trapezium with parallel sides 5 cm and 9 cm and height 4 cm.
\(\tfrac12bh\); \(bh\); \(\tfrac12(a+b)h\). Trapezium: \(\tfrac12\times14\times4=28\text{ cm}^2\). Use the perpendicular height.
Give the volume and curved surface area of a cylinder, and find the volume when r = 3 cm and h = 10 cm.
\(V=\pi r^2h\), curved surface \(=2\pi rh\). \(V=\pi\times9\times10=90\pi=283\text{ cm}^3\) (3 s.f.).
Give the cone and sphere formulae and find the volume of a sphere of radius 3 cm.
Cone: \(V=\tfrac13\pi r^2h\), curved area \(\pi rl\). Sphere: \(V=\tfrac43\pi r^3\), area \(4\pi r^2\). Sphere \(r=3\): \(36\pi=113\text{ cm}^3\).
How do you find the volume of a pyramid? A square-based pyramid has base 6 cm and height 5 cm.
\(V=\tfrac13\times\text{base area}\times\text{height}=\tfrac13\times36\times5=60\text{ cm}^3\).
Convert 2.5 m² into cm². Why is the factor not 100?
Area units scale as the square of the length factor: \(1\text{ m}^2=100^2=10\,000\text{ cm}^2\), so \(2.5\text{ m}^2=25\,000\text{ cm}^2\). Volumes use \(100^3\).
How do you construct the perpendicular bisector of a line AB?
Set the compasses to more than half of AB. Draw arcs from A and from B so they cross above and below the line, then join the two crossing points. Keep the construction arcs.
(Higher) Give the period and range of y = sin x, y = cos x and y = tan x.
\(\sin x\) and \(\cos x\) repeat every \(360^\circ\) with values between \(-1\) and 1. \(\tan x\) repeats every \(180^\circ\), takes all values, and has asymptotes at \(90^\circ\), \(270^\circ\), ...
(Higher) Solve sin x = 0.5 for 0° ≤ x ≤ 360°, and state the two key identities.
\(x=30^\circ\) and \(180-30=150^\circ\). Identities: \(\tan x=\dfrac{\sin x}{\cos x}\) and \(\sin^2x+\cos^2x=1\).

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