The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. It can be used to find the length of the third side of a right triangle if the other two sides are known. The theorem is commonly used in fields like architecture, engineering, and navigation to calculate distances, lengths, and other spatial relationships.

2. The theorem, also known as the Pythagoras
theorem,
It states that the
square of the length of the hypotenuse is equal to
the sum of squares of the lengths of other two
sides of the right-angled triangle.

3. a = side of right triangle
b = side of right triangle
c = hypotenuse

5. Why Is This Useful?
If we know the lengths of two sides
of a right angled triangle, we can
find the length of the third side.
(But remember it only works on
right angled triangles!)

6. example: What is the diagonal distance across a square
of size 1?
Unit Square Diagonal
Start with: a2 + b2 = c2
Put in what we know: 12 + 12 = c2
Calculate squares: 1 + 1 = c2
1+1=2: 2 = c2
Swap sides: c2 = 2
Square root of both sides: c = √2
Which is about: c = 1.4142...

7. Does this triangle have a Right Angle?
10 24 26 triangle
Does a² + b² = c² ?
a² + b² = 10²+ 24² = 100 + 576 = 676
c² = 26² = 676
They are equal, so ...
Yes, it does have a Right Angle!
Example: Does an 8, 15, 16 triangle have a Right
Angle?
Does 8 ²+ 15²= 16²?
8² + 15² = 64 + 225 = 289,
but 162 = 256
So, NO, it does not have a Right Angle

8. pythagorean Theorem Derivation
Consider a right angled triangleDelta ABC. It is right
angled at B.
In Delta ABC and Delta ADB
∠ABC=∠ABD=90∘
∠DBA=∠BCA
∠A=∠A
Using the AA criterion for the similarity of triangles,
ΔABC≅ΔADB
Considering ΔABCandΔBDC
∠DAB=∠CBD

9. ∠DBA=∠BCD
∠CDB=∠ADB=90∘
Using the AA criterion for the similarity of triangles,
ΔBDC≅ΔABC
Thus, it can be concluded ΔADB≅ΔBDC
So if a perpendicular is drawn from the right-angled
vertex of a right triangle to the hypotenuse, then the
triangles formed on both sides of the perpendicular are
similar to each other and also to the whole triangle.
Now, we are required to prove AC2=AB2+BC2. We
drop a perpendicular BD on the side AC.
We already know that ΔADB≅ΔABC

10. ADAB=ABBC (Condition for similarity) Or
AB2=AC×AD…….. (1)
Also, ΔBDC≅ΔABC
∴CDBC=BCAC (Condition for similarity) Or
BC2=AC×CD……….. (2)
Summing equation (1) with equation (2),
AB²+BC²=CD.AC+AD.AC
⇒AB²+BC²=AC(CD+AD)
However, CD+AD=AC
Thus, AC²=AB²+BC²

11. First construct a perpendicular from C to segment AB.
In a right triangle, the altitude from the right angle to
the hypotenuse divides the hypotenuse into two
segments (e and f). The length of each leg of the right
triangle is the geometric mean of the lengths of the
hypotenuse and the segment of the hypotenuse that is
adjacent to the leg, therefore
c/a=a/e and c/b=b/f
Ce=a² cf=b²

12. then adding a and b yields:
a²+b²=ce+cf
factoring out c on the right side gives:
a²+b²=c(e+f)
,
and looking at the above diagram:
c=e+f
so by substitution we get :
a²+b²=c²

13. Aerospace scientists and meteorologists find the range
and sound source using the Pythagoras theorem.
It is used by oceanographers to determine the speed of
sound in water.
Application of Pythagoras Theorem in Real Life:
Architecture and Construction.