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Pythagoras Theorem

Triangles II - Concepts

Calculating Perimeter of Triangle

Area of Similar Triangles

Similarity Related To Right Triangle

Pythagoras Theorem

Results Deduced From Pythagoras Theorem

Class - RRB Technician Grade I Signal Subjects

Concept Explanation

Pythagoras Theorem

Pythagoras Theorem:

Pythagoras, the famous Greek philosopher, lived about 572 B.C. to 501 B.C. He proved the relation between the lengths of the sides of a triangle, Although, this theorem was known to the Babylonians 1000 years ealier, but, Pythagoras is believed to have the first to discovered a proof of this theorem. However, long ago ( 800 B.C.), the Indian mathe,atician Baudhaya had stated and proved this property of a right angled triangle.

Let us have a look at the fig. $large Delta ABC$ is a right triangle, right angled at C, so that AB is the hypotenuse and AC and BC are the sides of the right triangle,

Then, $large (AB)^{2}=(BC)^{2}+(CA)^{2}$

i.e., $large (Hypotenuse)^{2}=(Base)^{2}+(perpendicular)^{2}$

Given: A triangle ABC right angled at B

To Prove: $large AB^2+BC^2=AC^2$

Construction: From B draw BD $large perp$ AC

Proof: Since BD $large perp$ AC

$large Delta ADBsimDelta ABC$ [By the above theorem]

$large Rightarrow frac{AD}{AB}=frac{AB}{AC}$

$large Rightarrow AB^2= AD;X;AC$ ............(1)

Now $large Delta BDCsimDelta ABC$ [By the above theorem]

$large Rightarrow frac{BC}{AC}=frac{DC}{BC}$

$large Rightarrow BC^2= AC;X;DC$ ............(2)

Adding Eq (1) and (2)

$large AB^2 +BC^2= AC;X;AD + AC;X;DC$

$large AB^2 + BC^2= AC;(AD + ;DC)$

$large AB^2 +BC^2= AC;X;AC$

$large AB^2 +BC^2= AC^2$

Hence Proved

Example 1: The lengths of two sides of a right triangle are 5 cm and 12 cm. Find the length of the hypotenuse.

Solution: Suppose BC = 5cm and AC = 12 cm.

$large therefore$ By Pythagoras theorem,

$large (AB)^{2}=(BC)^{2}+(AC)^{2}$

$large (AB)^{2}=(5)^{2}+(12)^{2}$

$large =25+144$

$large =169$

$large (AB)^{2}=169$

$large ABtimes AB=13times 13$

$large Rightarrow$ $large or;;AB=13 cm$

$large therefore$ Length of the hypotenuse = 13 cm.

Converse of Pythagoras Theorem:

THeorem: In a triangle if the square of one side is equal to the sum of squares of the other two sides, then the angle opposite to the first side is a right angle.

Given: A triangle ABC such that $large AB^2+ BC^2=AC^2$

To Prove : $large angle B =90^0$

Construction : Construct a triangle DEF such that DE= AB, EF= BC and $large angle E =90^0$

Proof: Since $large Delta DEF$ is a right angled triangle with right angle at E

$large therefore DF^2=DE^2+EF^2$

Since DE = AB and EF = BC replacing them in the above equation

$large therefore DF^2=AB^2+BC^2$

But $large AB^2+ BC^2=AC^2$ [Given]

$large therefore DF^2=AC^2$

DF = AC

Now in $large Delta$ ABC and $large Delta$ DEF

AB = DE [By Construction]

BC= EF [By Construction ]

DF = AC [Proved Above]

$large therefore Delta ABCcongDelta DEF$ [By SSS Congruence Criterion]

$large Rightarrow angle B = angle E=90^0$

Hence $large Delta$ ABC is right angled at B

Illustration: ABC is an isosceles triangle with Ac = BC. If $large AB^2=2AC^2$ . Prove that ABC is aright triangle.

In $large Delta$ ABC, we are given that

AC = BC and $large AB^2=2AC^2$

Now $large AC^2= BC^2$

Adding $large AC^2$ To both sides we get

$large AC^2+AC^2=BC^2+AC^2$

$large BC^2+AC^2=2AC^2$

But $large AB^2=2AC^2$

$large therefore BC^2+AC^2=AB^2$

From Converse of Pythagoras Theorem we can say that

Triangle ABC is a right angled at C

Sample Questions

(More Questions for each concept available in Login)

Question : 1

The length of the hypotenuse of a right triangle exceeds the length of the base by 2 cm and exceeds twice the length of the altitude by 1 cm. Find the length of each side of the triangle.
A. 5, 13, 17	B. 7, 11, 17	C. 8, 15, 17	D. 11, 15, 17
Right Option : C
View Explanation

Question : 2

A man goes 150 m due east and then 200 m due north. How far is he from the starting point?
A. 150 m	B. 250 m	C. 130 m	D. 130 m
Right Option : B
View Explanation

Question : 3

If square of one side of triangle is equal to the sum of the square of other two sides then triangle is ______________________.
A. Acute a triangle	B. Obtuse triangle	C. Right triangle	D. None of these
Right Option : C
View Explanation

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Pythagoras Theorem

Pythagoras Theorem

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