O is the centre of the circle having radius 5 cm. AB and AC are two chords such that AB = AC = 6 cm. If OA meets BC at M, then OM =

3.6 cm

No worries! We‘ve got your back. Try BYJU‘S free classes today!

1.4 cm

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

2 cm

No worries! We‘ve got your back. Try BYJU‘S free classes today!

3 cm

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is B 1.4 cm
$G i v e n - O i s t h e c e n t r e o f a c i r c l e . A B = A C = 6 c m a r e c h o r d s . A O i s j o i n e d a n d e x t e n d e d t o m e e t B C a t M . T o f i n d o u t - t h e l e n g t h o f O M = ? S o l u t i o n - W e j o i n O B & O C . W e c o n s i d e r f i g . I . L e t O M = x . i . e A M = A O + x = (5 + x) c m . A l s o l e t B M = y c m . N o w b e t w e e n Δ A O C & Δ A O B w e h a v e O A = O B = O C (r a d i i o f t h e s a m e c i r c l e) & A B = A C = 6 c m . ∴ B y S S S t e s t Δ A O C ≅ Δ A O B ⟹ ∠ O A C = ∠ O A B . A g a i n b e t w e e n Δ A M C & Δ A M B w e h a v e A B = A C, A M c o m m o n s i d e, ∠ O A C = ∠ O A B . ∴ Δ A M C ≅ Δ A M B ⟹ ∠ A M C = ∠ A M B = 90^{o} (l i n e a r p a i r) . S o Δ A M B i s a r i g h t o n e w i t h A B a s h y p o t e n u s e . ∴ B y P y t h a g o r a s t h e o r e m, w e g e t {A M}^{2} = {A B}^{2} - {B M}^{2} ⟹ {(5 + x)}^{2} = 6^{2} - y^{2} ⟹ x^{2} + y^{2} = 11 - 10 x . . . . . . (i) . A l s o Δ O M B i s a i s a r i g h t o n e w i t h O B a s h y p o t e n u s e . ∴ B y P y t h a g o r a s t h e o r e m, w e g e t {O M}^{2} = {O B}^{2} - {B M}^{2} ⟹ x^{2} = 5^{2} - y^{2} ⟹ x^{2} + y^{2} = 25...... (i) . . . . . . . . (i i) C o m p a r i n g (i) & (i i), w e h a v e 11 - 10 x = 25 ⟹ x = - 1.4 c m . T h e n e g a t i v e s i g n s h o w s A O > A M . S o w e c o n s i d e r f i g I I . H e r e A M = (5 - x) c m ∴ (i) b e c o m e s x^{2} + y^{2} = 11 + 10 x . S o, c o m p a r i n g t h i s e q u a t i o n w i t h (i i) a n d s o l v i n g s i m u l t e n e u s l y, w e g e t x = 1.4 c m . ∴ O M = 1.4 c m . a n s - O p t i o n B .$

Suggest Corrections

Similar questions

A B

A C

A B = A C = 6 c m

5 c m

B C

A B = A C = 6 c m

A B

A C

5

A B = A C = 4 \sqrt{5}

B C

A B = A C = 6

In a circle of radius 5 cm, $A B$ and $A C$ are two chords such that $A B = A C = 6 c m .$ Find the length of the chord $B C$ .

Join BYJU'S Learning Program