Intro To Geometry //

Question 1

5 axioms of Euclid

Answer 1

1. Through any two points there passes a unique line
2. It is possible to extend any line segment continuously in a straight line to a larger line segment
3. It is possible to draw a unique circle of any given radius around any given point
4. All right angles are equal to each other
5. (Parallel postulate) if a straight line crossing two straight lines makes interior angles on one of the sides which are less than two right angles in total, then the two straight lines, if produced indefinitely, meet on that side

Question 2

Angle from a to b

Answer 2

Given two intersecting lines a and b, we denote by ∠ab the value of either of the two equal angles where we go from a to b anti-clockwise

Question 3

Perpendicular

Answer 3

Two lines a and b are said to be perpendicular (a ⊥ b) if they intersect at right angles, that is ∠ab = ∠ba = π/2

Question 4

Map (geometry)

Answer 4

A map from the plane to the plane is a rule that sends each point in the plane to some other point in the plane. Given a map f and a point P in the plane, we denote f(P) the point where f sends P

Question 5

Perpendicular bisect

Answer 5

Given any line segment AB its perpendicular bisect is the unique line which passes through the midpoint of AB and is perpendicular to AB

Question 6

Reflection

Answer 6

The reflection r_l in l is the map which sends any point P to the unique point r_l(P) such that l is the perpendicular bisect of the line segment Pr_l(P)

Question 7

Dilation D_O,α

Answer 7

Given a point in the plane define the dilation D_O,α with centre O and scale factor α to be the map which leaves the point O fixed and sends any point P ≠ O to the unique point D_O,α(P) which lies on the continuation of the line segment OP in the direction of P at the distance α|OP| from O

Question 8

Translation T_v

Answer 8

Given a vector v = AB (oriented line segment), define the translation T_v by v to be the map which translates every point in the plane by AB

Question 9

Isometry

Answer 9

A map f: Plane → Plane is an isometry if for any two points A and B in the plane we have d(A,B) = d(f(A),f(B))

Question 10

Rotation R_O,θ

Answer 10

Given a point O in the plane the rotation R_O,θ with centre O and angle 0 ≤ θ < 2π is the map such that for every point P ≠ O, R_O,θ(P) is the unique point P’ on the circle with centre O and radius |OP| such that ∠POP’ = θ

Question 11

Orientation preserving

Answer 11

An isometry f: Plane → Plane is orientation preserving if for any three points A, B and C in the plane we have ∠ABC = ∠f(A)f(B)f(C)

Question 12

Orientation reversing

Answer 12

An isometry f: Plane → Plane is orientation reversing if for any three points A, B and C in the plane we have ∠ABC = ∠f(C)f(B)f(A)

Question 13

Congruent

Answer 13

Two geometrical figures are congruent if there exists an isometry taking one to the other, denoted by ≅. If a vertex order is specified it must be preserved, e.g. ΔABC ≅ ΔA’B’C’ requires that there exists an isometry taking A to A’, B to B’ and C to C’

Question 14

Clockwise orientated triangle

Answer 14

A triangle ΔABC is clockwise orientated if moving from A to B to C takes you clockwise around the points in the interior of the triangle

Question 15

3 criteria for congruence of ΔABC and ΔA’B’C’

Answer 15

1. (SAS) |BA|=|B’A’|, |BC|=|B’C’| and β=β’
2. (ASA) |BC|=|B’C’|, β=β’ and γ=γ’
3. (SSS) |AB|=|A’B’|, |BC|=|B’C’| and |CA|=|C’A’|

Question 16

Tangent

Answer 16

A line l is tangent to a circle C at point P if it meets C only at P

Question 17

Distance from P to ℓ

Answer 17

The distance from a point P to a line ℓ is 0 if P ∈ ℓ, and is |OP| for the unique O ∈ ℓ with PO ⊥ ℓ, otherwise

Question 18

Great circle/geodesic

Answer 18

A great circle on a sphere is the intersection of this sphere with a plane passing through its centre

Question 19

Antipodal

Answer 19

Two points on a sphere are called antipodal if they lie on opposite ends of a diameter of the sphere

Question 20

Ambient distance (3 space)

Answer 20

Distance from A to B is the length of the straight line joining them

Question 21

Intrinsic distance

Answer 21

The distance from A to B is the length of the smaller of the two arcs of any great circle joining A and B

Question 22

Tangent plane

Answer 22

Let S be the surface of a sphere in a 3D space, let O be its centre and P ∈ S a point. Then the plane through P, perpendicular to OP, is the unique plane tangent to S at P

Question 23

Spherical angle

Answer 23

Let AB and AC be two arcs of two great circles ℓ_B and ℓ_C on a sphere S. Let π_A be the plane tangent to S at A, and let t_B and t_C be the tangent lines carved out on π_A by the planes of ℓ_B and ℓ_C respectively. The spherical angle ∠BAC between ℓ_B and ℓ_C is defined to be the planar angle ∠t_B,t_C between t_B and t_C in plane π_A

Question 24

Parallel Postulate

Answer 24

If a straight line crossing two straight lines makes the interior angles on one of the sides which are less than two right angles in total, then the two straight lines, if produced indefinitely, meet on that side

Question 25

Playfair’s axiom

Answer 25

Given a line ℓ and a point P not on ℓ there is a unique line through P parallel to P

Question 26

3 axioms defining area

Answer 26

1. The area of a rectangle is the product of the lengths of its two adjacent sides
2. If two figures are disjoint or meet only along their edges, then the area of their union is the sum of their areas
3. Congruent figures have equal areas

Question 27

Pythagoras’ theorem

Answer 27

Let ΔABC be a triangle in the plane such that B is a right angle, then |AB|^2 + |BC|^2 = |AC|^2

Question 28

Sum of angles in a triangle ΔABC in a sphere

Answer 28

∠A + ∠B + ∠C = π + Area(ΔABC)/R², where R = radius of sphere

Question 29

Stereographic projection

Answer 29

A function F: S{p_n} → ∏, where ∀ A ∈ S, p_nA ⊂ E^3 is the unique straight line passing through A and p_n, and F(A) ∈ ∏ is the unique intersection point of p_nA and ∏

Question 30

Inversion wrt a circle C₀

Answer 30

The map I_C₀: ∏ \ {O} → ∏ \ {O}, defined by I_C₀(P) = F(refl_∏(F⁻¹(P))) for every P ∈ ∏ \ {O}, and refl_∏ is the reflection with respect to ∏