Question

$\bf{\boxed{\huge{\mathlab{HOLA \: MATES}}}}$

Class : 9

Subject : Maths

Chapter : Triangles

$\underline{\texit{\large{QUESTION :}}}$

If D is the midpoint of the hypotenuse AC of a right angled triangle ABC. Prove that :

$\bf{BD \: = \: 1/2 \: AC}$

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Answer 1

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Prove that:-
If D is the midpoint of the hypotenuse AC of a right angled triangle ABC. Prove that :


BAD=1/2AC


Given:-

If D is the midpoint of the hypotenuse AC of a right angled triangle ABC


AD = DC

Construction:-

BD = DE

And,
∠ADB = ∠CDE {Reason:- Vertically opposite Angle}


Using SAS criterion of congruence


∆ADB ≅ ∆CDE 

=) EC = AB
∠CED = ∠ABD ....(i)[Reason:- Cpct}

Here
∠CED and ∠ABD are alternate interior angles 

∴ CE ║ AB

∠ABC + ∠ECB = 180°[ReASoN;-
Consecutive interior angles]

=) 90 + ∠ECB = 180° 

⇒ ∠ECB = 90°

Now,

In ∆ABC and ∆ECB

AB = EC [By using Equation(i)] 

BC = BC [Reson:-Common] 

And,

∠ABC = ∠ECB = 90° 

∴ BY Using [ SAS criterion of congruence] 

∆ABC ≅ ∆ECB 

=) AC = EB [By cpctc] 

=} 1/2 AC = 1/2 EB 

=) BD = 1/2 AC 

$\huge{ \bold{ \fbox{ \color{green}{proved}}}}$