Pythagorean theorem

detailed discuss pythagorean theorem

1. Pythagorean theorem

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

4. 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!)

5. 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...

6. Does this triangle have a Right Angle? 10 24 26 triangle Does a ² + b ² = c ² ? a ² + b ² = 10 ² + 24 ² = 100 + 576 = 676 c ² = 26 ² = 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 + 15 ² = 16 ² ? 8 ² + 15 ² = 64 + 225 = 289, but 162 = 256 So, NO, it does not have a Right Angle

7. pythagorean Theorem Derivation Consider a right angled triangle\Delta 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, Δ ABC≅ Δ ADB Considering Δ ABCand Δ BDC ∠DAB=∠CBD

8. ∠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

9. ADAB=ABBC (Condition for similarity) Or AB2=AC×AD…….. (1) Also, Δ BDC≅ Δ ABC ∴CDBC=BCAC (Condition for similarity) Or BC2=AC×CD……….. (2) Summing equation (1) with equation (2), AB ² +BC ² =CD.AC+AD.AC ⇒ AB ² +BC ² =AC(CD+AD ) However, CD+AD=AC Thus, AC ² =AB ² +BC ²

10. given triangle ABC, prove that a² + b² = c². 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=a ² cf =b ²

11. then adding a and b yields : a ² +b ² = ce+cf factoring out c on the right side gives: a ² +b ² =c( e+f ) , and looking at the above diagram: c = e+f so by substitution we get : a ² +b ² =c ²

12. 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.