state and prove pythagoras
Answers
Pythagorean Theorem
If c is the hypotenuse and sides are a and b, the right triangle satisfies a² + b² = c².
Einstein's Proof
(Refer to the attachment.)
Einstein proved it with three triangles.
ΔABC, ΔBCH, ΔACH are similar. Let's rotate the triangles.
We see the similarity ratio is c : a : b. So the areas will be fc², fa², fb² respectively.
- ΔABC = fc²
- ΔBCH = fa²
- ΔACH = fb²
He noted the area of triangles because ΔABC is the sum of two triangles.
So, fa² + fb² = fc².
Divide by f, then we get a² + b² = c².
Extra information
The converse of the Pythagorean theorem is true: If a² + b² = c², it is a right triangle.
The contraposition is true also: If a² + b² ≠ c², it is not a right triangle.
Mathematical words
A condition p: ΔABC is a right triangle.
A condition q: a² + b² = c²
Pythagorean theorem: p then q.
*converse: q then p. Not always true.
*contraposition: not q then not p. Always true.
Statement: In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Given: ABC is a triangle in which ∠ABC=90∘
Construction: Draw BD⊥AC.
Proof:
In △ADB and △ABC
∠A=∠A [Common angle]
∠ADB=∠ABC [Each 90∘]
△ADB∼△ABC [A−A Criteria]
So, ABAD=ACAB
Now, AB2=AD×AC ..........(1)
Similarly,
BC2=CD×AC ..........(2)
Adding equations (1) and (2) we get,
AB2+BC2=AD×AC+CD×AC
=AC(AD+CD)
=AC×AC
∴AB2+BC2=AC2 [henceproved]