It was noon at The Temple when Miss Loi was totally engrossed in trying to obtain the proof in a nasty Plane Geometry question sent earlier by a student.
After staring for what seemed like an eternity at the question’s diagram, her 阴阳眼 simply would not come and she was beginning to get very tired, partly due to the ridiculous phone call that cut short her sleep in the morning.
Never one to give up easily in the face of adversity and face the ignominy of hearing a “Haha! Miss Loi also dunno how to do!“, she rubbed her eyes and continued to stare intensely at the diagram.
So intense was her gaze that a circle in the diagram seemed to grow larger and larger as it beckoned her to lean closer and closer … so close that she felt almost at one with the question and could almost feel the closeness of the solution that she desperately sought.
Suddenly, an unknown force reached out and, to her horror, pulled her headfirst into the circle!
In the harrowing moments that followed, she was whisked through a tunnel lined with circles, triangles, tangents, secants, chords and all manner of grotesque polygons from the O-Level A-Maths Syllabus.
After what seemed like an eternity, she landed with a big thump right in front of a weathered-looking Right-Angled Triangle (RAT), who muttered
Oh boy here comes another 走火入魔 one.
Who are you? Where am I?
asked Miss Loi as she meticulously checked for any facial scar arising from the rough landing.
You’re at the Plane of Geometry. The realm of all killer Plane Geometry questions. And you, of all people, should know that I’m the Right-Angled Triangle alias Pythagoras Theorem – introduced to most during Sec One/Two and some say even PSLE! But they hardly think about me anymore.
Can you show me the way back to The Temple?
The RAT continued to grumble.
These days all the young people here only care about the ‘latest’ fads like the Tangent-Secant Theorem etc. They hardly think about me anymore.
Uncle! I really need your help! I’ve got a class to teach in a few hours’ time! AND I WANT MY NATIONAL DAY LONG WEEKEND!!!
I will show you the way if you can show me the following:
P is a point outside the circle such that PA and PB are tangents to the circle of centre O. M is the mid-point of AB and the secant line PDC cuts the chord AB at E.
Show that:
- ΔPMA is similar to ΔBMO
- PA×BM = BO×PM
- PE² = PD×PC − DE×CE
As Miss Loi has somehow lost her 阴阳眼 ability, can you help her find her way back to The Temple?
Meanwhile the RAT continued to mumble …
They hardly think about me anymore …
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Miss Loi is a full-time private tutor in Singapore specializing in O-Level Maths tuition. Her life's calling is to eradicate the terrifying LMBFH Syndrome off the face of this planet. For over 21 years she has been a savior to countless students ... 
曜
日
Tora! Tora! Tora!
(13)click to see how horribly dirty it is
With the return of the haze and Miss Loi’s recent absence away in the Plane of Geometry, the poor car is looking a little neglected these days.
Dismayed and feeling depressed like a guilty parent each time she saw the dismal state of its paintwork, Miss Loi decided to send the car for a long-overdue wash plus pamper it with one of those much-touted “car grooming” sessions.
When she collected her car later in the day, her depression was cured instantly by the uplifting sight of its dazzling shine.
Words could not describe the silky sensation as she ran her slim fingers slowly across its smooth immaculately-waxed body, now free of the minuscule scratches and parabolic swirls that once scarred it.
It was almost like getting a new car, and this feeling stayed with her throughout her drive to a new student’s home, situated at the end of a straight road in a carpark-deprived area, for her next tuition session.
- Miss Loi’s somewhere down there
As she slowly walked away, turning her head every few steps to admire the shine of her parked car, little did she realize that this glint of a shine would attract the attention of a little bird perched on a large tree at the end of the road.
Before long the area was slowly draped by a large shadow that caused Miss Loi to stop in her tracks, look up, and stare in utter horror at a huge fleet of birds flying towards her direction in an aggressive-looking formation, each with its … umm … “crosshairs” trained on the car.
The air raid siren in her mind sounded, and it was sheer luck that got her to the car in time before these terrors in the sky rained destruction upon the land.
In heroic scenes that followed, left and right she swerved her car as she strived to save it from the relentless onslaught, narrowly avoiding bomb after bomb of waste matter that whistled through the air before landing with a sickening *splat* inches away from her.
Just when it seemed certain the car would be martyred on this very day, the flock began to turn back, as Miss Loi charged her way to freedom beyond the bombing range, immensely relieved that her superb driving skills had somehow saved her from the pounding.
But now she’s stuck at the end of the road, in a tense face-off with the evil birds on the tree at the opposite end readying themselves for the next bombing run, while her new student waits patiently for her in her house …
Noticing the pattern of the impact points along the road, Miss Loi suspects that their distance d from the centre of the road and the distance x along the road is connected by the equation
Bravely measuring a few samples along certain intervals on the road, Miss Loi recorded the following sets of values:
Given there’s nowhere else she can park her car other than along the centre of the narrow road (inside the bombing range), *use a suitable graph to examine the validity of Miss Loi’s equation, and estimate the values of a and b.
At which point (x) along the road should Miss Loi park her car to minimize the chances of it being hit i.e. where d is maximum?
As Miss Loi doesn’t have any graph paper with her (shame on her!), can you please help her find the safest spot to park her shiny car so that she can start her new student’s tuition session on time with peace of mind?