Pythagoras theorem
Pythagoras Theorem is also called the Pythagorean theorem. This is named after a Greek mathematician called Pythagoras. This theorem explains the relationship of the sides of the right angle triangle. the side of the right angle triangle also called as Pythagorean triples.
Pythagoras Theorem statement:
Pythagoras Theorem states that “ in a right angle triangle the square of the hypotenuse side is equal to the sum of the squares of the other two sides ”.
The three sides of the triangle are hypotenuse, base and perpendicular. Hypotenuse is the longest side of the triangle or it can also be said as the side opposite to 90.
Pythagoras Theorem proof :
Given : A right angle triangle ABC, right angled at B.
To prove: (AC)² = (AB)² + (BC)²
Construction: Draw a perpendicular line BD to meet AC at D.
In a right angle triangle ABC AB = 6 BC = 8 find AC
Converse of Pythagoras theorem
Converse of Pythagoras Theorem statement:
The Converse of Pythagoras Theorem states that “ if the square of a side is equal to the sum of the squares of the other two sides then the triangle is a right angled triangle”
Converse of Pythagoras Theorem proof:
Given: (AC)² = (AB)² + (BC)²
To prove : ABC is a right angled triangle.
Construction: construct a triangle EFG such that AB = EF = A , BC = FG = b
Examples of Pythagoras theorem:
Find the length of AE?
ABCE is a rectangle, AB=EC = 6m
BC = AC = 12m
Applying Pythagoras theorem, we get,
(AD)²= (DE)²+ (AE)²
(AD)² = 122 + 52
(AD)² = 144 + 25
(AD)² = 169
AD = 13 m
Find the perimeter of the rectangle, whose length is 24cm and diagonal is 26cm?
Let us take BCD
AC=24 , DB = Hypotenus = 26
Using Pythagoras theorem,
DB2 = DC2 +CB2
(26)² = (24)² + (CB)²
676 = 576 + (CB)²
676 - 576 = (CB)²
100 = (CB)²
CB = 10
L=24 ; B= 10
Perimeter of a rectangle = 2( L+B)
= 2( 24 + 10 )
= 2 ( 34)
= 68
A ladder 8.5m long rests against a vertical wall its foot 4m away from the wall. How high up the wall the ladder reach?
Using Pythagoras theorem,
(AC)² = (BC)² + (AB)²
(8.5)²= (BC)² + 42
72.25 - 16 = (BC)²
56.25 = (BC)²
BC = 7.5m
Therefore the ladder reaches the wall 7.5m
Examples of Pythagoras Theorem :
The top of a ladder of length 15 m reaches a window 9 m above the ground. What is the distance between the base of the wall and that of the ladder ?
Let LN be a ladder of length 15 m that is resting against a wall. Let M be the base of the wall and L be the position of the window.
The window is 9 m above the ground. Now, MN is the distance between the base of the wall and that of the ladder.
In the right angled triangle LMN, ∠M = 90∘. Hence, side LN is the hypotenuse.
According to Pythagoras' theorem,
(LN)² = (MN)² + (LM)²
(15)² = (MN)² + (9)²
(225) = (MN)² + 81
(MN)² = 225 − 81
(MN)² = 144
(MN)² = (12)²
MN = 12
∴ Length of seg MN = 16 m.
Hence, the distance between the base2 of the wall and that of the ladder is 12 m.
In the right-angled ∆LMN, ∠ M = 90°. If l(LM) = 12 cm and l(LN) = 20 cm, find the length of seg MN.
In the right angled triangle LMN, ∠M = 90∘. Hence, side LN is the hypotenuse.
According to Pythagoras' theorem,
(LN)² = (MN)² + (LM)²
(20)² = (MN)² + (12)²
400 = (MN)² + 144
(MN)² = 400 − 144
(MN)² = 256
(MN)² = (16)²
(MN) = 16
∴ Length of seg MN = 16 cm.
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