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The 阴阳眼 Of Plane Geometry

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Tuition given in the topic of A-Maths Tuition Notes & Tips from the tuition centre desk of Miss Loi at 8:34 pm (Singapore time)

A long time ago (okay not that long) in an old A-Maths Syllabus far, far away, there was an evil topic called Relative Velocity – where a mere mention of its name would strike fear into the hearts of students, caused teachers to quit, and made babies cry at night.

So one can imagine the national euphoria that reached epic proportions when the said topic was removed from the New Syllabus

… only to be replaced by another fiendish nightmare called Plane Geometry (or Geometric Proofs to some) that’s causing widespread pandemonium for students across the nation, making teachers stutter in class, and … umm … the babies haven’t stopped crying.

While it’s easy to say that everything you need to know is summarized in the following (along with this Similarity & Congruency chart):

Geometric Formulae For Plane Geometry

Go on. Click to download and print this out.

The reality, however, is quite often that mortals like you and me will probably stare and stare at the Plane Geometry question till we end up like this:

Stressed StudentStressed Student II

Crazyhamster & DK, don’t mind Miss Loi borrow your handsome faces for illustrative purposes ok?

But we suppose that a ‘killer’ topic like this is meant to separate the haves from the have-nots in that cumulative frquency curve, for only those on the brink of their Mathematical Nirvana will get to experience that life-changing moment of sudden realization, when you reach your Mathematical high, and your knees begin to tremble as you feel the power of the Midpoint/Intercept Theorem flowing through your body as you look up to see a glowing image of Miss Loi amongst the clouds in the sky.

Mathematical Nirvana

They have just found the proof in their Plane Geometry question.

If you desire to join Stephen Chow, Cristiano Ronaldo and whoever’s that Miss Universe in these emotive Plane Geometry moments, you must develop the 阴阳眼 of Plane Geometry, for the signs within those complex geometrical diagrams will only reveal themselves magically to those who possess it, and only then will you be able to ‘see’ eerie things that others don’t.

Contrary to the myth that one can only be born with it (as suspected in most foreign cyborgs cases), the 阴阳眼 (lit. Yin Yang Eye) can actually be acquired through long hours of old-fashioned practice.

To that end, Miss Loi has toiled through the week to string together the following series of Plane Geometry questions that she sincerely hope can help improve your ‘eyesight’ in time for your O-Level A-Maths papers this week.

THE PLANE GEOMETRY EYESIGHT TEST

Warning PLEASE DO NOT CLICK ON THE ANSWERS BEFORE YOU’VE ATTEMPTED THE QUESTION! You will NEVER be able to develop the 阴阳眼 by merely reading through the answers!

For a start, don’t always insist on jumping in with the latest theorems. Sometimes all you need are just the basic geometrical properties you’ve learnt in your primary school and Sec One Maths (see part 1 of this question).

Also when you see an obvious lack of circles in the diagram, be prepared to seek out and use the Intercept Theorem or the Midpoint Theorem at some point.

Triangle Diagram

In the given figure, RT = RQ. If SR bisects ∠PRQ,

  1. Prove that ∠PRS = ∠RTQ.
  2. Show that RP/RQ = SP/SQ.

[+] Warning ANSWER – you shall suffer a terrible fate if you click and peek at the answer before you’ve tried the question!

Answer Diagram

  1. Let ∠RTQ = ∠RQT = θ (∠s of isosceles Δ)
    ⇒ ∠PRQ = 2θ (ext. ∠ = sum of interior opposite ∠s of Δ)

    Since SQ bisects ∠PRQ,
    ⇒ ∠PRS = ∠QRS = θ
    ⇒ ∠PRS = ∠RTQ

  2. And since ∠PRS = ∠RTQ
    RS // TQ (corresponding ∠s)

    Using Intercept Theorem,
    RP/RT = SP/SQ doubleright RP/RQ = SP/SQ (∵ RQ = RT)

Instead of folding your arms and stare and stare at the diagram till you turn into stone, things can magically appear when you finally decide to lift up that 100kg pencil of yours to draw a couple of lines to join up all points in the diagram, for fortune tends to favour those who can lift and use their pencils.

Question 1 Diagram

Given a quadrilateral HIJK, let L, M, N and O denote the respective midpoints. Show that LMNO is a parallelogram.


[+] Warning ANSWER – you shall suffer a terrible fate if you click and peek at the answer before you’ve tried the question!

Answer 1 Diagram

MN // MN // LO (Midpoint Theorem)
NO // IK // ML (Midpoint Theorem)
LMNO is a parallelogram (∵ MN // LO & NO // ML

Ditto for this next question. You’ll be amazed how much 天机 (Heaven’s Will) is revealed by a single (correct) stroke of fate. Moreover, you’ll also discover that most Plane Geometry solutions are pretty short (typically 1-2 steps) since each part usually carries only 3-5 marks.

It’s the 阴阳眼 that you need!

Circle 1 Diagram

The diagram shows a circle with centre O and a chord AB. P is on AB and C is on the circle such that ∠OPC = 90°.

Prove PC2 = PAPB


[+] Warning ANSWER – you shall suffer a terrible fate if you click and peek at the answer before you’ve tried the question!

Answer Diagram

Extending the line PC to C’,
C’P = PC (Perpendicular bisector of chord)
C’PPC = PAPB (Intersecting Chords Theorem)
PC2 = PAPB (∵ C’P = PC)

Once again, join up the points dear – join up those points!

Circle 3 Diagram

ABCD is a cyclic quadrilateral. CE is a tangent to the circle at C. Given that AD // BC, prove that

  1. BCA = ∠DCE
  2. ABC = ∠BCD

[+] Warning ANSWER – you shall suffer a terrible fate if you click and peek at the answer before you’ve tried the question!

Answer Diagram

  1. DCE = ∠DBC (Alternate Segment Theorem)
    DBC = ∠DAC (∠s in same segment)
    DAC = ∠BCA (alternate ∠s)
    ⇒ ∠BCA = ∠DCE

  2. Let ∠DCE = x
    ⇒ ∠DBC = x (Alternate Segment Theorem)
    ⇒ ∠ACB = x (alternate ∠s)
    Let ∠ABD = y
    ⇒ ∠ACD = y (∠s in same segment)

    ∴ ∠ABC = x + y = ∠BCD

Have your eyes warmed up sufficiently by now? Can you now see ‘them’ now in the more complicated diagram below? Yes, they’re everywhere, crying out to you.

Question Diagram ABT is a tangent to circle PBCQ. The circles ABCD and PBCQ intersect at B and C. APC and BQD are straight lines intersecting at X.

  1. Prove that PQ is parallel to AD.
  2. Show that XP = {AB^2~XQ}/{AC~DQ}.

[+] Warning ANSWER – you shall suffer a terrible fate if you click and peek at the answer before you’ve tried the question!

  1. Answer 1 Diagram

    ACB = ∠ADB (∠s in same segment)
    PQB = ∠ACB (∠s in same segment)
    ⇒ ∠ADB = ∠PQB (corresponding ∠s)

    PQ // AD

  2. Answer 2 Diagram

    Since PQ // AD,
    XP/PA = XQ/DQ (Intercept Theorem)
    doubleright XP = PA {XQ/DQ}

    Also,
    ACPA = AB2 (Tangent Secant Theorem)
    doubleright PA = AB^2/AC
    XP = {AB^2~XQ}/{AC~DQ}

And now one of the so-called ‘killer’ question to wrap things up.

Question 2 Diagram ABCD is a rectangle whose vertices lie on the circumference of a circle, centre O. X is the midpoint of AB, and AD = AX. Y lies on BD and XYC is a straight line. Z lies on the circumference of the circle and DXZ is a straight line. AC and BD intersect at O. The line CZ meets AB at P.

  1. Prove that ΔDXY is similar to ΔDZB.
  2. Explain why a circle that passes through A, P and Z can be drawn.

[+] Warning ANSWER – you shall suffer a terrible fate if you click and peek at the answer before you’ve tried the question!

  1. Answer 2a Diagram

    ΔDXY and ΔDZB share ∠ZDB.
    DZB = 90° (∠ in semi-circle)
    AXD = ∠BXC = (180°-90°)/2 = 45° (interior ∠s of isosceles Δ)
    ⇒ ∠DXC = 180°-45°-45° = 90° (adjacent ∠s)
    ⇒ ∠DXC = DZB
    ⇒ ∠DYX = DBZ (sum of interior ∠s of Δ)
    ⇒ ΔDXY similar to ΔDZB (AAA)

  2. Answer 2b Diagram

    AZP = 180°-∠ADC (=90°) = 90° (Opposite ∠s of cyclic quadrilateral)
    ⇒ a circle that passes through A, P and Z (∵∠AZP is ∠ in semi-circle)

Most of the theorems have been covered till this point. And if you’ve successfully proven all the questions ON YOUR OWN, your knees will tremble as you receive that almighty gift of the 阴阳眼 Of Plane Geometry, knowing that no Plane Geometry question can stop you from now on.

If you don’t, you’ll just have to try the above questions all over again till you do 😉

Most importantly to all O-Level A-Maths students, may you find your 阴阳眼 in your A-Maths Papers this week!

GOOD LUCK!!!

P.S. Little Miss Loi wishes all Science students good luck for your O-Levels too!

頑張って!!!

Revision Exercise

To show that you have understood what Miss Loi just taught you from the centre, you must:

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Comments & Reactions

23 Comments

  1. Wendy's Avatar
    Wendy commented in tuition class


    2008
    Oct
    20
    Mon
    1:14am
     
    1

    HAR it got taken out!? (is part of the "long time ago", cough. it REALLY wasn't that long ago. really. REALLY!)

    Young people nowadays are so lucky!

  2. Miss Loi's Avatar
    Miss Loi Friend Miss Loi on Facebook @MissLoi commented in tuition class


    2008
    Oct
    20
    Mon
    9:36am
     
    2

    Yes, yes Wendy hence that "(okay not that long)" coz we'd all agree that it REALLY wasn't that long ago - in fact it was just last year 😉

    But think most Physics students would still prefer Relative Velocity leh.

  3. ash's Avatar
    ash commented in tuition class


    2008
    Oct
    21
    Tue
    12:17am
     
    3

    i is just the thing i am looking for.
    my weakest chapters in amath is plane geometry and sketching of graphs. they are killing me.

  4. Lisa's Avatar
    Lisa commented in tuition class


    2008
    Oct
    23
    Thu
    5:50pm
     
    4

    I just went through the whole thing... and I couldn't get a single one T_T. This type of plane geometry is x10 harder than the p.g. in the United States =( *sob*

  5. rispwaker's Avatar
    rispwaker commented in tuition class


    2008
    Oct
    24
    Fri
    1:07pm
     
    5

    Don't worry Lisa ! Just 1hr + away from Amaths Paper II and I can't do any of the questions either =(. But don't worry, I pray that God will grant us mercy and grace, to let us have that sudden 阴阳眼 for the particular question later on.

    =D

  6. Miss Loi's Avatar
    Miss Loi Friend Miss Loi on Facebook @MissLoi commented in tuition class


    2008
    Oct
    25
    Sat
    8:53pm
     
    6

    ash + rispwaker: Were you gifted the 阴阳眼 that allowed you to 'see' that eerie Mid-Point Theorem triangle on Friday?

    Lisa: You sure these are 10x harder than those in the US? Miss Loi has just received some sample Geometric Proof questions from Hong Kong and oh my ... o.O Think AMaths students there require 阴阳 Lasik surgery 😛

  7. rispwaker's Avatar
    rispwaker commented in tuition class


    2008
    Oct
    25
    Sat
    9:28pm
     
    7

    *cough... I did. I was was tossing and turning, swirling and revolving the entire paper, in hope of that 阴阳眼 would come.

    And before I knew... an invigilator came stomping towards me. I could sense impeding doom, and just as he stood before me. I looked at her, then looked back at my paper, and OH GOSH! I SAW THE MID-POINT THEOREM ! It must be because of her solemn yet suggestive stare at me, that the 阴阳眼 popped out.

    Well, but knowing is one thing, explaining is another. I just merely wrote some nonsense, then suddenly, pop in a midpoint theorem ( I did not know how to prove that mid-point theorem was valid), in hope that the old and foggy cambridge markers would just look at the GOLDEN *Mid-Point Theorem* and give me that valuable 5 marks.

  8. mido's Avatar
    mido commented in tuition class


    2008
    Oct
    25
    Sat
    10:04pm
     
    8

    hmm i thought it was similar triangles to solve that mid point problem.

    one triangle has radius as a side. one triangle has diameter as a side. prove ratio 2:1.

  9. Anoneemers's Avatar
    Anoneemers commented in tuition class


    2008
    Oct
    25
    Sat
    10:14pm
     
    9

    I saw the mid-point theorem after 10 minutes of and staring at the paper... The paper was at 2.30pm and i slept at 1am the night before so my 阴阳眼 cannot activate immediately =.=
    My steps are proving 2 lines parallel using corresponding angle and then since O is midpoint, E is midpoint...is it correct?

  10. rispwaker's Avatar
    rispwaker commented in tuition class


    2008
    Oct
    25
    Sat
    10:54pm
     
    10

    Well, no one knows for sure, until MISS Loi makes her holy appearance and show us the light out of this question.

  11. Anoneemers's Avatar
    Anoneemers commented in tuition class


    2008
    Oct
    25
    Sat
    11:02pm
     
    11

    p.s. i forgot to type the word so my sentence sounds weird.... The 1st statement should read: I saw the mid-point theorem after 10 minutes of MEDITATION and staring at the paper

  12. Lisa's Avatar
    Lisa commented in tuition class


    2008
    Oct
    26
    Sun
    12:33am
     
    12

    Yeah Miss Loi, the plane geometry learned here includes theorems my teachers never covered before (e.g. alternate segment theorem, midpoint theorem, intercept theorem) in the US. We don't call it plane geometry in the US though... the topic is called proofs. We need to write out two columns, Statements & Reasons... then prove what is asked, similar to this topic. It also involves circles and triangles... One thing easier about S'pore proofs is that it's much easier writing it out after you've figured out how to solve the problem. In the US, you write LOADS for your proof. If you have a long problem, it might take half a page... writing out... things such as...
    ABC is a right angle - Perpendicular lines intersect to form right angles
    DEF is a right angle - Perpendicular lines intersect to form right angles
    ABC is equal to DEF - All right angles are equal
    Lameee =.=

    and rispwaker, I'm not taking the O-levels this year haha... I just moved from the US a few weeks ago here and I'm hoping to enroll in a good JC in Singapore =_=. So in order to know what I should already know going into JCs, I should completely master O-level math (which I am failing). ;(

  13. rispwaker's Avatar
    rispwaker commented in tuition class


    2008
    Oct
    26
    Sun
    12:37am
     
    13

    I think it would be quite challenging to find a JC here, since you just came from United States. ( Education here is tough. No joke. )

    But anyways, all the best to you. <3

  14. Miss Loi's Avatar
    Miss Loi Friend Miss Loi on Facebook @MissLoi commented in tuition class


    2008
    Oct
    26
    Sun
    2:41am
     
    14

    Alright everyone, by popular request here's Miss Loi's solution to the inaugural Plane Geometry aka Geometric Proof question of the GCE O' Level 2008 AMaths Paper II.

    *DENG DENG DENG DENG ...*

    First, the question:

    GCE O-Level 2008 Plane Geometry Question

    The diagram shows a circle, centre O, with diameter AB. The point C lies on the circle. The tangent to the circle at A meets BC extended at D. The tangent to the circle at C meets the line AD at E.

    1. Prove that triangles AEO and CEO are congruent.
    2. Prove that E is the mid-point of AD.

    [+] Warning ANSWER - The same rule applies here: you shall suffer a terrible fate if you click and peek at the answer before you've tried the question!

    Answer to Part i.

    GCE O-Level 2008 Plane Geometry Answer Part i

    OE (Common side)
    OA = OC (Radius of circle)
    AE = CE (Tangents from external point)
    EAO = ECO = 90° (Radius ⊥ tangent)

    Just pick the relevant ones from the pasar malam of points above to prove that ΔAEO ≡ ΔCEO via SSS or SAS case!

    Now for the answer to Part ii:

    GCE O-Level 2008 Plane Geometry Answer Part ii

    *Summons the mighty 阴阳眼*

    Since O is the mid-point of AB, so if E is the mid-point of AD, then like what Anoneemers said all we need is to prove OE // BD for the Mid-point Theorem to hold.

    Let ∠ABC be θ.

    *Summons the mighty 阴阳眼 again *

    Looking at the arc AC, ∠AOC = 2θ (∠ at centre = 2 x ∠ at circumference)
    ⇒ ∠AOE = θAEO ≡ ΔCEO)
    OE // BD (∵ ∠AOE = ∠ABC → corresponding ∠s)

    So by the Mid-point Theorem, E is the mid-point of AD. YAY!


    But do note that the above is by no means the only way to prove that E is the mid-point of AD.

    Like what mido said, you can also prove that ΔAEO and ΔADB are similar, and thereby using the ratio:

    AO/AB = 1/2 = AE/AD

    to deduce that E is the mid-point of AD. But Miss Loi's fingers are a little too tired now to type out the proof for the similarity part 😛

    Anyone with more otherworldly 阴阳眼 experiences to share from Friday???

    P.S. Miss Loi hopes that rispwaker's 阴阳眼 on Friday was really zooming in on the AMaths paper and not at some suggestive spot on the invigilator.

  15. Anoneemers's Avatar
    Anoneemers commented in tuition class


    2008
    Oct
    26
    Sun
    3:22pm
     
    15

    There's another 阴阳眼 question on Friday which is the 9(i) proving L=4sinθ-2cosθ (p.s. this is not plane geometry) and alot of people dunno how to prove because they see right angles on 2 sides and then they thought is kite =.= then i was thinking if its a kite then L=2cm which is impossible...
    So i drew many many lines n eventually got the answer

  16. crazyhamster's Avatar
    crazyhamster commented in tuition class


    2008
    Dec
    9
    Tue
    1:02am
  17. Miss Loi's Avatar
    Miss Loi Friend Miss Loi on Facebook @MissLoi commented in tuition class


    2008
    Dec
    10
    Wed
    1:07am
     
    17

    Anoneemers: Yes there should be more than one way to prove L in that kite-ish diagram in 9(i). Did your 阴阳眼 saw the same stuffs as Miss Loi's?

    Crazyhamster: Believe it or not, it was found via searching for "stressed student" in Google Images!

    Google may have picked you as the iconic symbol of all stressed Singaporean students!

  18. Calvin's Avatar
    Calvin commented in tuition class


    2008
    Dec
    28
    Sun
    11:44am
     
    18

    Err okay maybe this is a little late to say this, but there seems to be a small typo with question 1...

    The question reads: "In the given figure, RT = RQ. If SQ bisects ∠PRQ" when it kinda looks more like SR bisecting ∠PRQ...

    Part 2:
    The question asks to show that (RT/RQ) = (SP/SQ)... but isn't RT = RQ?

    I dunno, maybe I'm just hallucinating the mistakes or something :S

  19. Miss Loi's Avatar
    Miss Loi Friend Miss Loi on Facebook @MissLoi commented in tuition class


    2008
    Dec
    28
    Sun
    12:04pm
     
    19

    Thanks Calvin for pointing out the typos.

    Yes in Qn 1, SR should be the angle bisector and the said ratio in the question has been amended to RP/RQ.

    And it's not you but Miss Loi who was actually hallucinating when she was typing this whole chunk of swollen-fingers-inducing article many weeks ago 😛

    Do ring the bell loud loud again should you spot any more typos.

  20. Calvin's Avatar
    Calvin commented in tuition class


    2008
    Dec
    28
    Sun
    12:44pm
     
    20

    Okay. So my vision's still working perfectly 😛

  21. JH's Avatar
    JH commented in tuition class


    2009
    Mar
    30
    Mon
    10:18pm
     
    21

    Erm for the 4th question,is it possible to conclude that
    angle ABC=angle BCD=90 degree by using the right angle in semi circle theorem?

  22. Miss Loi's Avatar
    Miss Loi Friend Miss Loi on Facebook @MissLoi commented in tuition class


    2009
    Mar
    30
    Mon
    11:26pm
     
    22

    JH: For your conclusion to be true, AC & BD have to be diameters of the circle in order for ABC & BCD to be semi-circles, but this is not given in the question.

    Also note that though CE is tangent to the circle at C, it's not stated in the question that AC is ⊥ to CE, so we can't assume AC to be a diameter using the radius ⊥ tangent property either.

    However, do share with Miss Loi any alternative method should your own 阴阳眼 happens to reveal something different 😉

  23. Jukie Primestein's Avatar
    Jukie Primestein commented in tuition class


    2010
    Oct
    30
    Sat
    4:27am
     
    23

    Miss Loi I like the way u express your proof with diagrams. Thanks for the geometry formula sheet I have never seen one. With the diagrams of both I will be able to teach geometry even to the youths who can't read.

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