MyQuestionIcon

1

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Direction Cosines

If AD, BE a...

Question

If $A D, B E a n d C F$ be medians of a $Δ A B C$ ,then $2 (A D + B E + C F) < k (A B + B C + C A) < 4 (A D + B E + C F)$ .Find k

Open in App

Solution

$G i v e n : I n △ A B C, A D, B E a n d C F a r e m e d i a n s . I n △ A B C, A D, B E a n d C F a r e m e d i a n s . W e k n o w t h a t t h e s e a r e c o n c u r r e n t . L e t t h e s e m e e t a t G . A l s o A G : G D = B G : G E = C G : G F = 2 : 1 S o, A G = \frac{2}{3} A D, B G = \frac{2}{3} B E, C G = \frac{2}{3} C F . . . . . (1) B e c a u s e i n t r i a n g l e, t h e s u m o f t w o s i d e s i s g r e a t e r t h a n t h e t h i r d, S o, i n △ A G B, A G + B G > A B I n △ B G C, B G + C G > B C I n △ C G A, C G + A G > C A A d d i n g t h e s e, w e g e t 2 (A G + B G + C G) > A B + B C + C A \Rightarrow 2 (\frac{2}{3} A D + \frac{2}{3} B E + \frac{2}{3} C F) > A B + B C + C A [F r o m (1)] \Rightarrow \frac{4}{3} (A D + B E + C F) > A B + B C + C A \Rightarrow 4 (A D + B E + C F) > 3 (A B + B C + C A) . . . . (2) N o w i n △ A B D A B + B D > A D \Rightarrow A B + \frac{1}{2} B C > A D (s i n c e A D i s a m e d i a n) S i m i l a r l y, B C + \frac{1}{2} C A > B E a n d C A + \frac{1}{2} A B > C F A d d i n g t h e s e, w e g e t \frac{3}{2} (A B + B C + C A) > A D + B E + C F \Rightarrow 3 (A B + B C + C A) > 2 (A D + B E + C F) . . . . . . (3) C o m b i n i n g (2) a n d (3), w e g e t 2 (A D + B E + C F) < 3 (A B + B C + C A) < 4 (A D + B E + C F) . H e n c e P r o v e d$

Suggest Corrections

thumbs-up

0

Similar questions

△ A B C

A D, B E

C F

intersect each other at

G

4 (A D + B E + C F) > 3 (A B + B C + C A)

Q. AD, BE and CF are the medians in ΔABC. If AD + BE + CF = a and AB + BC + CA = b, then which of the following is always correct?

Δ A B C

A D, B E a n d C F

are the altitudes. If

A B \times A C = 28.80

B E \times C F = 20

A D \times B C

A D, B E

C F

are the medians of a

△ A B C

(A D^{2} + B E^{2} + C F^{2}) : (B C^{2} + C A^{2} + A B^{2})

A D, B E

C F

are the medians of

Δ A B C,

then the value of

(A D^{2} + B E^{2} + C F^{2}) : (B C^{2} + C A^{2} + A B^{2})

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

Direction Cosines and Direction Ratios

MATHEMATICS

Watch in App

explore_more_icon

Explore more

Direction Cosines

Standard XII Mathematics

Join BYJU'S Learning Program

CrossIcon