Geometry Pythagorean Theorem

Geometry Pythagorean theorem states that the square of hypotenuse is equal to the sum of the squares of other two legs.
Given : A right angled triangle ABC in which ∠C = 900
Prove that : c2 = a2 + b2
Construction : From C draw CD ⊥ BC
Statements
Reasons
1) ∠ADC = ∠ACB 1) Each 900 and by construction
2) ∠A = ∠A 2) Reflexive (common)
3) ΔADC ~ ΔABC 3) AA similarity
4) AD / AC = AC / AB 4) If two triangles are similar then corresponding sides are proportional
5) AC2 = AD x AB 5) Cross multiplication
6) ∠BDC = ∠ACB 6) Each 900 and by construction
7) ∠B = ∠B 7) Reflexive (common)
8) ΔCDB = ΔABC 8) AA similarity
9)BD / BC = BC / AB If two triangles are similar then corresponding sides are proportional
10) BC2 = BD x AB 10) Cross multiplication
11) AC2 + BC2 =AD x AB + BD x AB 11) Add (5) and (10)
12) AC2 + BC2 = AB ( AD + BD ) 12) Taking AB as common
13) AC2 + BC2 = AB x AB 13) Substitution ( AD + BD = AB)
14) AC2 + BC2 = AB2 14) Simplify
15) c2 = a2 + b2 15) From the diagram. Hence proved.

Note : The converse of this theorem is also true.

If the square of hypotenuse is equal to the sum of squares of other two sides then it is a right angled triangle.
If c2 = a2 + b2 the ABC is a right triangle at C.
• Geometry proofs
• GeometryProof-1
• GeometryProof 2
• area-similartriangles
• geometry pythagorean theorem

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