Proof of Pythagorean Theorem
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
Now, we are sure that square WXYZ = square EFGH + 4 ∆ GYF
or, (b + c)2 = a2 + 4 ∙ 1/2 b ∙ c
or, b2 + c2 +
or, b2 + c2 = a2
Proof of Pythagorean Theorem using Algebra:
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)
Conditions for the Congruence of Triangles
Right Angle Hypotenuse Side congruence
Converse of Pythagorean Theorem
7th Grade Math Problems
8th Grade Math Practice
From Proof of Pythagorean Theorem to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
Recent Articles
-
Multiplying 3-Digit Number by 1-Digit Number | Three-Digit Multiplicat
Oct 22, 24 03:26 PM
Here we will learn multiplying 3-digit number by 1-digit number. In two different ways we will learn to multiply a two-digit number by a one-digit number. 1. Multiply 201 by 3 Step I: Arrange the numb… -
Word Problems on Multiplication |Multiplication Word Problem Worksheet
Oct 22, 24 01:23 AM
Word problems on multiplication for fourth grade students are solved here step by step. Problem Sums Involving Multiplication: 1. 24 folders each has 56 sheets of paper inside them. How many sheets of… -
Worksheet on Word Problems on Multiplication | Multiplication Problems
Oct 22, 24 12:31 AM
In worksheet on word problems on multiplication, all grade students can practice the questions on word problems involving multiplication. This exercise sheet on word problems on multiplication -
Multiplying 2-Digit Number by 1-Digit Number | Multiply Two-Digit Numb
Oct 21, 24 03:38 PM
Here we will learn multiplying 2-digit number by 1-digit number. In two different ways we will learn to multiply a two-digit number by a one-digit number. Examples of multiplying 2-digit number by -
Multiplication Table of 4 |Read and Write the Table of 4|4 Times Table
Oct 21, 24 02:26 AM
Repeated addition by 4’s means the multiplication table of 4. (i) When 5 candle-stands having four candles each. By repeated addition we can show 4 + 4 + 4 + 4 + 4 = 20 Then, four 5 times
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.