in a equilateral triangle ABC D is a point on side BC such that BD is equal to 1 by 3 BC prove that 9ad
square is equal to 7 a b square
Answers
Answer:
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Step-by-step explanation:
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Here is your solution
Given:-
ABC is an equilateral triangle.
D is point on BC .
so BD =BC.
To prove:-
9 AD² = 7 AB²
Construction: Draw AE ⊥ BC.
Proof ;-
Considering on Triangles which are given below;-
In a ΔABC and ΔACE
AB = AC ( given)
AE = AE (common)
∠AEB = ∠AEC = (Right angle)
∴ ΔABC ≅ ΔACE
By RHS Creation
∴ ΔABC ≅ ΔACE
Again,
BE = EC (By C.P.C.T)
BE = EC = BC²
In a right angled ΔADE
AD²= AE2 + DE² ---(1)
In a right angled ΔABE
AB² = AE² + BE² ---(2)
From equation (1) and (2) ;
=) AD² - AB² = DE² - BE².
=) AD² - AB² = (BE – BD)² - BE².
= ) AD² - AB² = (BC / 2 – BC/3)² – (BC/2)²
= AD2 - AB2 = ((3BC – 2BC/6)² – (BC/2)²
= AD² - AB² = (BC² / 36 – BC2 / 4 )
( In a equilateral triangle, All sides are equal to each other)
AB = BC = AC
= ) AD²= AB² + AB²/ 36 – AB² / 4
= )AD² = (36AB² + AB²– 9AB²) / 36
= ) AD² = (28AB²) / 36
=) AD² = (7AB²) / 9
Cross Multiplication here,
= ) 9AD² = 7AB²
Hence,
9AD² = 7AB² proved
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