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. 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,
i.e.,
Given: A triangle ABC right angled at B To Prove: Construction: From B draw BD AC Proof: Since BD AC [By the above theorem]
............(1) Now [By the above theorem]
............(2) Adding Eq (1) and (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.
By Pythagoras theorem,
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 To Prove : Construction : Construct a triangle DEF such that DE= AB, EF= BC and Proof: Since is a right angled triangle with right angle at E Since DE = AB and EF = BC replacing them in the above equation But [Given] DF = AC Now in ABC and DEF AB = DE [By Construction] BC= EF [By Construction ] DF = AC [Proved Above] [By SSS Congruence Criterion] Hence ABC is right angled at B |
Illustration: ABC is an isosceles triangle with Ac = BC. If . Prove that ABC is aright triangle.
In ABC, we are given that AC = BC and Now Adding To both sides we get But From Converse of Pythagoras Theorem we can say that Triangle ABC is a right angled at C |
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. | |||
Right Option : C | |||
View Explanation |
In fig. length of AE is ________________ | |||
Right Option : C | |||
View Explanation |
In a right angle triangle ABC, right angled at B , AB = 6cm , BC = 8cm , then AC = _________________ . | |||
Right Option : A | |||
View Explanation |
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