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90 = 100: a proof

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A (purported) geometric proof that 90=100.

Construct a four-sided figure ABED as follows:

  • |ABE| = 90
  • |DEB| = 100
  • |AB| = |ED|
Using that as a starting point, we now tinker a bit to show that 90=100:
  • Draw the perpendicular bisectors to BE and AD; call the point where they meetC.
Actually, we must prove that those two perpendicular bisectors really do meet at all ( i.e. , that the point C even exists). In this case, it turns out to be pretty clearit's not hard to argue that lines AD and BE aren't parallel,and therefore their perpendicular bisectors aren't parallel, and so they must intersect (in Euclidean geometry).Still, be alert for people making glib assertions in proofs.

A construction to help prove that 90=100

Looking at this figure, some warning flags should be going up: How do we know C lies below BD? Might it lie above BD? Or exactly on BD?It turns out that the argument below is the same in all of these cases, though you'll certainly want to verify this to yourself later.

1 |AB| = |ED| By construction.
2 |BC| = |EC| C is on the perpendicular bisector of BE (thus△BEC is isosceles).
3 CBEBEC Base angles of isosceles triangle BEC are congruent.
4 |CBE| = |BEC| Congruent angles have equal measures; line 3.
5 |AC| = |DC| C is on the perpendicular bisector of AD (thus△ADC is isosceles).
6 △ABC△DEC (!!) Triangles with three congruent sides are congruent (Euclid's Side-Side-Side congruence theorem); lines 1,2,5.
7 (From here, it's just routine steps to conclude 90=100:)
8 ABCDEC Corresponding parts of congruent triangles are congruent; line 6.
9 |ABC| = |DEC| Congruent angles have equal measures; line 8.
10 |ABC| = |ABE| + |CBE| By construction.
11 |DEC| = |DEB| + |BEC| By construction.
12 |DEC| = |DEB| + |CBE| Substituting equals with equals; lines 11 and 4.
13 |ABC| = |DEB| + |CBE| Substituting equals with equals; lines 12 and 9.
14 |ABE| + |CBE| = |DEB| + |CBE| Substituting equals with equals; lines 13 and 10.
15 |ABE| = |DEB| Subtracting equals from equals remains equal.
16 90 = |DEB| By construction, and substituting equals with equals; line 15.
17 90 = 100 By construction, and substituting equals with equals; line 16.

A useful corollary: 0=1.

1 90 = 100 Previous theorem.
2 0 = 10 Subtracting equals (90) from equals remains equal.
3 0 = 1 Dividing equals by non-zero equals (10) remains equal.

If you feel this result is incorrect,then the challenge for you is to find the first line which is false.

The flaw is extremely hard to find. We won't actually give the solution, but here's a hint onhow to go about attacking the puzzle:

Note that finding the bug in the proof is the same skill as debugging a program.A good approach is to try various degenerate inputs. In this case, there are a couple ofinputsto the constructionthe length of CD is arbitrary; no matter how long or short the proof should apply equally well.Similarly, the angle 100seems arbitrary; fiddling with inputs like these (making them very small or very large)might give you some clues as to where the bug is. A very careful drawing will clear things up.

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You may have noticed that the proof given here has some very minuscule steps e.g. Congruent angles have equal measure.Usually such simple steps can be omitted, since they are obvious to any reader. We include them for a few reasons:

  • As a careful thinker, you should recognize that such small steps really are part of the complete reasoning,even if they're not worth mentioning continually.
  • If a computer is checking a proof, it needs to actually include those steps.
  • Programmers do need to be concerned with distinctions about (abstract) typesthe difference between angles and their measures, in this case.
  • Sometimes a line's justification is glibly given asby construction, when that may not even be correct !-) .
In this course, we'll spend a few weeks working with proofs which do include all the small, pedantic steps, to instill a mental framework for what a rigorous proof is.But after that, you can relax your proofs to leave out such low-levelsteps, once you appreciate that they are being omitted.

Questions & Answers

what does the ideal gas law states
Joy Reply
Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
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what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
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definitely of physics
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what is field
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physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
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Another formula for Acceleration
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Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
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No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
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Source:  OpenStax, Intro to logic. OpenStax CNX. Jan 29, 2008 Download for free at http://cnx.org/content/col10154/1.20
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