## Tag Archives: Foundation Maths

### Proof of SAS Congruency Test of two triangles

Theorem: If two triangles have two sides of the one equal to two sides of the other, each to each, and the angles included by those sides equal, then the triangles are equal in all respects.

Construction: Let ABC, DEF be two triangles in which $AB=DE$ and

$AC=DF$ and the included angle $\angle BAC$ is equal to included angle $\angle EDF$.

It is required to prove that the

$\Delta ABC = \Delta DEF$ in all respects.

Proof: Apply the $\Delta ABC$ to the $\Delta DEF$ so that the point A falls on the point D; and the side AB along the side DE. Then, because $AB=DE$ so the point B must coincide with the point E. And, because AB falls along DE, and the

$\angle BAC=\angle EDF$, so AC must fall along DF. And, because

$AC=DF$, the point C must coincide with the point F. Then, since B coincides with E, and C with F, hence, the side BC must coincide with the side EF. Hence, the $\Delta ABC$ coincides with the $\Delta DEF$, and is therefore equal to it in all respects. QED.

Nalin Pithwa

### Math Basics A Fallacy in Geometry

You might think that crazy things like 1=2 can be *proved* if we make basic blunders in Algebra. But, in geometry also you can *prove* wild things like a right angle is an obtuse angle — errr…hmm…provided you make flaws in some fundamental assumption/axiom/theorem/property. So, check how strong are your basics in Euclidean/plane or high school geometry — point out my mistake in the proof “a right angle is an obtuse angle!”.

More later,

Nalin

### From scratch — the 3 medians of a triangle are concurrent

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Please download the attachments, pages 1 and 2 respectively of the proof that the 3 medians of a triangle are concurrent. That’s called the power of axioms!! This is the example which mesmerized Albert Einstein as a child to Mathematics and later to  Physics.

More later…

Nalin

### Math from scratch

Math from scratch!

Math differs from all other pure sciences in the sense that it can be developed from *scratch*. In math jargon, it is called “to develop from first principles”. You might see such questions in Calculus and also Physics.

This is called Axiomatic-Deductive Logic and was first seen in the works of Euclid’s Elements (Plane Geometry) about 2500 years ago. The ability to think from “first principles” can be developed in high-school with proper understanding and practise of Euclid’s Geometry.

For example, as a child Albert Einstein was captivated to see a proof from *scratch* in Euclid’s Geometry that “the three medians of a triangle are concurrent”. (This, of course, does not need anyone to *verify* by drawing thousands of triangles and their mediansJ) That hooked the child Albert Einstein to Math, and later on to Physics.

An axiom is a statement which means “self-evident truth”. We accept axioms at face-value. So, there are axioms, definitions, propositions, lemmas, theorems and corollaries.

Euclid’s geometry rests on the following fundamental axioms:

1)      There can be one and only one straight line joining two given points.

2)      (a) If O is a point in a straight line AB, then a line OC, which turns about O from the position OA to the  position OB must pass through one position, and only one,, in which it is perpendicular to AB.

(b) All right angles are equal.

3) (a) If a point O moves from A to B along the straight line AB, it must pass through one         position in which it divides AB into two equal parts.

3)(b) If a line  OP, revolving about O, turns from OA to OB, it must pass through one position in which it divides the angle AOB into two equal parts.

4) Magnitudes which can be made to coincide with one another are equal.

5) Playfair’s Axiom: Through a given point, there can be only one straight line parallel to a given straight line.

Note that these are the only basic assumptions to be used in geometric constructions also with ruler and compass.

PS: article reblogged and slightly modified:

Reference: To start geometry from scratch, you can start working from the first page of a classic text ” A School Geometry” by Hall and Stevens, Metric Edition; For example, Amazon India link is:

https://www.amazon.in/School-Geometry-H-S-Hall/dp/9385923331/ref=sr_1_4?crid=6QK90ZAHHFAU&keywords=a+school+geometry+hall+and+stevens&qid=1561783333&s=books&sprefix=A+School%2Caps%2C253&sr=1-4

I would like to add a few more details as there are some students/readers who want to pursue this further. {By the way, I have used the above reference only. One more thing,…Dover publications still prints/publishes/sells the original volumes of Euclids books}:

$\textbf{Hypothetical Constructions}$

From the above axioms, it follows that we may suppose:

i) A straight line can be drawn perpendicular to a given straight line from any point in it.

ii) A finite straight line (that is, a segment) can be bisected.

iii) Any angle can be bisected by a line (we call such a line its angle bisector).

$\textbf{Superposition and Equality}$

AXIOM: Magnitudes which can be made to coincide with one another are equal.

This axiom implies that any line, angle, or figure may be taken up from its position, and without change in size or form, laid down upon a second line, angle, or figure, for the purpose of comparison, and ti states that two such magnitudes are equal when one can be exactly placed over the other without overlapping.

This process is called superposition, and the first magnitude is said to be applied to the other. (Note: this is the essence of “congruency” relation in geometry).

$\textbf{Postulates}$

In order to draw geometric figures, certain instruments are required. These are — a straight ruler, and a pair of compasses. The following postulates (or requests) claim the use of these instruments, and assume that with their help the processes mentioned below may be duly performed:

Let it be granted:

1. That a straight line may be drawn from any one point to any other point.
2.  That a finite (or terminated) straight line may be produced (that is, prolonged) to any length in that straight line.
3. That a circle may be drawn with any point as centre and with a radius of any length.
4. some notes: Postulate 3 above implies that we may adjust the compasses to the length of any straight line PQ, and with a radius of this length draw a circle with any point O as centre. That is to say, the compasses may be used to transfer distances from one part of a diagram to another.
5. Hence, from AB, (a given terminated line), the greater of two straight lines, we may cut off a part equal to PQ the less. Because, if with centre A, and radius equal to PQ, we draw an arc of a circle cutting AB at X, it is obvious that AX is equal to PQ.

More later,

Regards,

Nalin Pithwa.