proof of pythagoras theorem
Answers
Answer:
In a right angle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. States that in a right triangle that, the square of a (a2) plus the square of b (b2) is equal to the square of c (c2)
The proof of Pythagorean Theorem in mathematics is very important.
In a right angle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
States that in a right triangle that, the square of a (a2) plus the square of b (b2) is equal to the square of c (c2).
In short it is written as: a2 + b2 = c2
Let QR = a, RP = b and PQ = c. Now, draw a square WXYZ of side (b + c). Take points E, F, G, H on sides WX, XY, YZ and ZW respectively such that WE = XF = YG = ZH = b.
Then, we will get 4 right-angled triangle, hypotenuse of each of them is ‘a’: remaining sides of each of them are band c. Remaining part of the figure is the
square EFGH, each of whose side is a, so area of the square EFGH is a2.
Now, we are sure that square WXYZ = square EFGH + 4 ∆ GYF
or, (b + c)2 = a2 + 4 ∙ 1/2 b ∙ c
or, b2 + c2 + 2bc = a2 + 2bc
or, b2 + c2 = a2
Proof of Pythagorean Theorem using Algebra:
Proof of Pythagorean TheoremGiven: A ∆ XYZ in which ∠XYZ = 90°.
To prove: XZ2 = XY2 + YZ2
Construction: Draw YO ⊥ XZ
Proof: In ∆XOY and ∆XYZ, we have,
∠X = ∠X → common
∠XOY = ∠XYZ → each equal to 90°
Therefore, ∆ XOY ~ ∆ XYZ → by AA-similarity
⇒ XO/XY = XY/XZ
⇒ XO × XZ = XY2 ----------------- (i)
In ∆YOZ and ∆XYZ, we have,
∠Z = ∠Z → common
∠YOZ = ∠XYZ → each equal to 90°
Therefore, ∆ YOZ ~ ∆ XYZ → by AA-similarity
⇒ OZ/YZ = YZ/XZ
⇒ OZ × XZ = YZ2 ----------------- (ii)
From (i) and (ii) we get,
XO × XZ + OZ × XZ = (XY2 + YZ2)
⇒ (XO + OZ) × XZ = (XY2 + YZ2)
⇒ XZ × XZ = (XY2 + YZ2)
⇒ XZ 2 = (XY2 + YZ2)
Step-by-step explanation:
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