1) In triangle ABC, AD is the median to side BC Prove that AB+AC is greater than 2AD - Maths - Triangles

Question

1) In triangle ABC, AD is the median to side BC Prove that AB+AC
is greater than 2AD
2)Prove that a line bisecting the vertical angle of a triangle
is perpendicular to the base the triangle is isosceles

Priyanka Kedia · Accepted Answer

$(1) G i v e n : I n Δ A B C, A D i s m e d i a n T o \Pr o v e : A B + A C > 2 A D C o n s t r u c t i o n : \Pr o d u c e A D t o E s u c h t h a t A D = D E a n d t h e n j o i n E C$

$\Pr o o f : I n Δ A D B a n d Δ E D C A D = D E (b y c o n s t r u c t i o n) ∠ A D B = ∠ C D E (v e r t i c a l l y o p p o s i t e a n g l e s a r e e q u a l) B D = D C (A D i s m e d i a n, s o D i s t h e m i d p o int o f B C) T h u s, b y S A S c r i t e r i o n Δ A D B ≅ Δ E D C S o, A B = E C (1) A n d, i n Δ A E C, A C + E C > A E (s u m o f a n y t w o s i d e s o f a t r i a n g l e i s g r e a t e r t h a n t h e t h i r d s i d e) \Rightarrow A C + A B > A E (A B = E C, f r o m (1)) \Rightarrow A C + A B > A D + D E \Rightarrow A C + A B > A D + A D \Rightarrow A C + A B > 2 A D h e n c e p r o v e d$ $(2) L e t A B C b e a t r i a n g l e a n d l i n e b i \sec t i n g t h e v e r t i c a l a n g l e o f a t r i a n g l e i s p e r p e n d i c u l a r t o t h e b a s e o f a t r i a n g l e S o, A D i s m e d i a n a n d A D ⊥ B C T o \Pr o v e : Δ A B C i s i s o s c e l e s$

$\Pr o o f : I n Δ A B D a n d Δ A D C, A D = A D (c o m m o n l i n e) ∠ A D B = ∠ A D C (e a c h h a s 90^{0} a n g l e) B D = D C (A D i s m e d i a n, s o D i s m i d p o int o f B C) T h u s, Δ A B D ≅ Δ A D C S o, A B = A C (c o r r e s p o n d i n g p a r t s o f c o n g r u e n t t r i a n g l e a r e e q u a l) \Rightarrow Δ A B C i s i s o s c e l e s h e n c e p r o v e d$

1) In triangle ABC, AD is the median to side BC. Prove that AB+AC is greater than 2AD. 2)Prove that a line bisecting the vertical angle of a triangle is perpendicular to the base the triangle is isosceles.

1) In triangle ABC, AD is the median to side BC. Prove that AB+AC is greater than 2AD.

2)Prove that a line bisecting the vertical angle of a triangle is perpendicular to the base the triangle is isosceles.