The Pythagorean Theorem and Its Converse?

Question

The Pythagorean Converse: You can take a rope and tie 12 equally spaced knots in it. You can then use the rope to check that a corner is a right angle. Why does this methos work?

Answer 1

A(squared) + B(squared) = C (squared)

Its converse?

C(squared) - B(squared) = A (squared)
or
C(squared)- A(squared) = B squared.

Answer 2

This is NOT sa converse -- this is an inverse. 12 units can be divided into 3, 4, and 5. so holding the rope at the appropriate knots against two sides that meet at a point will confirm that the rope fits snugly, with hypotenuse of 5. However if you hold the wrong knots you cannot confirm the result as a right angle.
It is NOT a good way, for several reasons: knots are not fine enough to serve as "point" vertices of a triangle, so the accuracy will be low (you could probably not tell the difference between 85 degrees and 90 degrees. Furthermore any triangle whose integral sides total to 12 could fit the rope. The nearest is the triangle 4 x 4 x 4 which identifies a 60 degree angle which is significantly different from 90 degrees, but that is "accidental": if you tried it with 30 knots, this would form a 5x12x13 Pythagorean triangle, but 5 x 11 x 14 forms a NON-Pythagorean triangle with one angle that may "look" like 90 degees to the untrained eye.

Answer 3

Given that 12 equally-spaced knots are tied in 1 rope, the rope has been divided into 12 equal segments. You can form a triangle with this rope by forming an angle at each interval of 3 knots, 4 knots, and 5 knots. You now have a right triangle, proven by the Pythagorean Theorem.

Pythagorean Theorem: Given a triangle: a and b are the legs; c is the hypotenuse.
a squared + b squared = c squared
In a right triangle, the sum of the squares of the legs equals the square of the hypotenuse.
Therefore: 3 squared + 4 squared = 5 squared
9 + 16 = 25
25 = 25

Proof of the Converse: The rope forms a right angle because the triangle it forms can be proven to be a right triangle by the Pythagorean Theorem.

Answer 4

Pythagoras's Theorem states that the square of the hypotenuse in a right angled triangle is equal to the sum of the squares of the other two sides. The classic right angled triangle has sides of 3, 4 and 5 so:-

3 x 3 = 9
4 x 4 = 16
Add 9 and 16 = 25
5 x 5 = 25

Now then 3 = 4 = 5 = 12 so if you make a right angle anywhere on the knots on the rope, according to your calculations (by the way you'd need 13 knots as you need 12 spaces.) you should be able to prove it. However if one side is 11 that leaves you with 1 and nothing for the hypotenuse so the rope theory doesn't work. It will for a 3, 4, 5 triangle though.

Answer 5

WTF, sound to me like people have the wrong notion of what the converse is.of a statement.

Dude, you answer can be found in symbolic logic (which is used to done proofs in math, which is why it helps if you look at the proof of whatever it is you are trying to understand.)

Here is some basic logic.
If I have a statement a=>b (a implies b, if a then b, a arrow b)
then the
converse is b=>a
inverse is ~a=>~b
contrapositive is ~b=>~a

~ symbol means "not" like the negation in English

Notice that the inverse is just converse contrapositive or the converse is the inverse contrapositive. And also out of all of these the contrapositive is the most important because the contrapositive has the property of being exactly equal to the original statement. We use this fact when we are doing proofs by contradiction which makes it really easy.

Okay, here is your answer. Out of all four of these, if the statement is true(or false), then the ONLY the contrapositive is guaranteed to be true(or false). You have no idea about the other two. You cannot say if the inverse is also true (or false) or if the converse is true (or false).

Now, it turns out that you can only say something about the converse and the inverse if the original statement a=>b is actually a<=>b (a double-arrow b, a if and only if b, a equivalent to b).

If you know that the statement a<=>b is true, then the inverse, converse, and the contrapositive are all true.

That's why the converse to the pythogorean theorem works. Because the statements
a=the triangle is a right triangle
b=one leg^2 + other leg^2 = hypotenuse ^2
were proven to be equivalent.

So I can assume a and prove b or I can assume b and prove a. Which means I can assume a and b will always be true or I can assume b and a will always be true.

Answer 6

The reason the converse works is because of the properties of congruent and similar triangles.

Remember in geometry class when you defined congruent triangles as those with three pairs of corresponding sides congruent and three pairs of corresponding angles congruent? Then you learned this happy short-cut called "SSS," or the side-side-side theorem?

Imagine a 3:4:5 triangle. You know this triangle to be a right triangle. Because of SSS, any other triangle that measures 3:4:5 (of the same unit) has to be congruent, and therefore must have a right angle opposite from the hypotenuse.

With similar triangles, the pairs of corresponding angles must all be congruent. The lengths of the sides of one triangle may differ, but they'll be in proportion to the corresponding sides of the other triangle.

Take your looped rope. Measure 12 equally-spaced knots in it. Pull out the sides in proportion to 3:4:5 knots. This triangle, no matter how far apart the knots are spread, is in proportion to any 3:4:5 triangle. They make similar triangles, which means the angles on each must be congruent.

Therefore, regardless of the space between each of the knots on your rope, as long as the sides of the triangle are in a 3:4:5 ratio, it will have a right angle opposite the longest side.

Answer 7

3, 4, 5 is a pythagorean triple (numbers that satisfy the pythagorean theorem)
3+4+5=12 which means if you connect the rope from end to end, you'll form a 3-4-5 right triangle.

Answer 8

a^2+ b^2=c^2
3+4+5=12
3^2+ 4^2= 5^2
9+16=25

Answer 9

Well, all bulding sites use the 3, 4, 5. They get a triangle with one side 3 feet, and one 4 feet and the final one five and that's a perfect right angle. We can note that 3, 4, and 5 make 12; so that's why it works.

Answer 10

12 = 3 + 4 + 5... 3^2 + 4^2 = 5^2

so one right triangle measures 3,4,5... if you take your rope, put 3 knots on one wall to the corner, 4 knots along the other wall... if it is 5 knots back out in open air.. then the wall's corner is square.

Answer 11

Once you believe the Pythagorean theorem, its converse follows very nicely in Euclid's Elements, Book I, Proposition 48. You should check it out, those books are famous for a reason!