Section 1 - Straight-edge and Compass Constructions

Question 1

How do you draw a circle from 3 non-colinear points?

Answer 1

• draw 2 lines AB & BC
• bisect them, midpt of AB is D & midpt of BC is E
• draw perp. to AB & BC at midpts
• these intersect at F
• claim F is centre of circle
• △ADF = △BDF by SAS
• AF = BF = CF
• radius = BF

Question 2

What is the Gauss-Wentzel thm?

Answer 2

a regular n-gon can be constructed w/ straight-edge & compass iff n is a product of a power of 2 and zero or more Fermat primes.

Question 3

what are Fermat primes?

Answer 3

primes of the form 2^(2^a) +1

Question 4

what do 𝙸.2 and 𝙸.3 tell us?

Answer 4

we can use a compass to store length

Question 5

what does 𝙸.4 tell us?

Answer 5

SAS

Question 6

what does 𝙸.8 tell us?

Answer 6

SSS

Question 7

what does 𝙸.26 tell us?

Answer 7

ASA & SAA

Question 8

how do you construct a regular 4-gon on a given base?

Answer 8

• AB is given
• draw perp. to AB at A & perp. to AB at B
• extend the perps. D&C and measure so AB=AD=BC
• draw CD
• then DC = AB and int. angles are right angles

construction is proved using isosceles triangles and 𝙸.28 & 𝙸.29
or using a circle with diameter CA

Question 9

what does 𝙸.28 state?

Answer 9

If a straight line falling on two straight lines makes the exterior angle equal to the interior and opposite angle on the same side (or the sum of the interior angles on the same side equal to two right angles) then the straight lines are parallel to one another.

Question 10

what does 𝙸.29 state?

Answer 10

A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the sum of the interior angles on the same side equal to two right angles.

Question 11

how do you divide a given line segment into n equal parts?

Answer 11

• given line AB
• draw l (not equal to AB) through A
• measure with compass so |AA₁|=|A₁A₂|=…=|Aₙ﹣₁Aₙ| on l
• draw BAₙ and parallels through Aₙ﹣₁ etc…
• these lines intersect AB and divide it into n equal parts

Question 12

what does Thales’ thm state?

Answer 12

parallels cut any line they cross into proportional segments

Question 13

when is a number constructible?

Answer 13

if a line segment of that length can be constructed from one unit length by straight-edge and compass

Question 14

if a & b are constructible then…

Answer 14

so are a+b, a-b (if a>b), ab, and b/a

Question 15

how do you construct ab?

Answer 15

• given a & b
• construct △0uA where |0u|=1 & |0a|=a
• extend 0u by b to get pt B
• draw parallels to uA at B
• let C be the pt where it meets the extension of 0A
• by Thales’ 1/a = b/|AC| so |AC|=ab

Question 16

how do you construct a/b?

Answer 16

• given a & b
• construct △0uA where |0u|=1 & |0a|=a
• extend 0A by b to get pt B
• draw parallels to Au at B
• let D be the pt where it meets the extension of 0u
• by Thales’ 1/a = |uD|/b so |uD|=d/a

Question 17

how do we know that any rational number is constructible?

Answer 17

pythagoras implies that if a&b are constructible the so is √(a²+b²)

Question 18

two triangles are similar if …..

Answer 18

… corresponding angles are equal

Question 19

how do we prove that similar triangles have corresponding proportional sides?

Answer 19

• imagine moving △A’B’C’ on top of △ABC so A’=A
• AC lies along A’C’ & AB lies along A’B’
• BC & B’C’ are parallel since they make the same angle with A’C’
• set |AC|=b, |CC’|=b’-b, |AB|=c, |BB’| = c’-c
• by Thales’ b/c = (b’-b)/(c’-c) an we obtain b/c = b’/c’ as desired
• thus the sides opposite ∠AB’C’ and ∠AC’B’ are proportional
• through the same process with the other angles we obtain the other 2 ratios needed