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

Byju's Answer

Standard IX

Mathematics

SSS Rule

In right tria...

Question

In right triangle ABC, right angled at C, M is the mid-point of hypotenuse AB. C is joined to M and produced to a point D such that DM = CM. Point D is joined to point B (see the given figure). Show that:
$∠ D B C i s a r i g h t a n g l e .$

Open in App

Solution

$I n Δ A M C a n d Δ B M D A M = B M (M i s m i d p o i n t o f A B) ∠ A M C = ∠ B M D (v e r t i c a l l y o p p o s i t e a n g l e s) C M = D M (g i v e n) ∴ Δ A M C ≅ Δ B M D (b y S A S c o n g r u e n c e r u l e) ∴ A C = B D (b y C P C T) A n d ∠ A C M = ∠ B D M (b y C P C T) W e h a v e ∠ A C M = ∠ B D M B u t A C M a n d B D M a r e a l t e r n a t e i n t e r i o r a n g l e s S i n c e a l t e r n a t e a n g l e s a r e e q u a l . H e n c e, w e c a n s a y t h a t D B | | A C \Rightarrow ∠ D B C + ∠ A C B = 180^{\circ} (c o - i n t e r i o r a n g l e s) \Rightarrow ∠ D B C + 90^{\circ} = 180^{\circ} (∠ A C B = 90^{\circ}) \Rightarrow ∠ D B C = 180^{\circ} - 90^{\circ} \Rightarrow ∠ D B C = 90^{\circ}$

Suggest Corrections

Similar questions

Q. In right triangle ABC, right angled at C, M is the mid-point of hypotenuse AB. C is joined to M and produced to a point D such that DM = CM. Point D is joined to point B (see the given figure). Show that:

Δ D B C ≅ Δ A C B

Q. Question 8 (ii)
In right triangle ABC, right angled at C, M is the mid-point of hypotenuse AB. C is joined to M and produced to a point D such that DM = CM. Point D is joined to point B (see the given figure). Show that:

∠ D B C i s a r i g h t a n g l e .

In right triangle $A B C$ , right angled at $C$ , $M$ is the mid-point of hypotenuse $A B$ . $C$ is joined to $M$ and produced to a point $D$ such that $D M = C M$ . Point $D$ is joined to point $B$ (see the given figure). Show that:

(i) $Δ A M C ≅ Δ B M D$

(ii) $∠ D B C$ is a right angle.

(iii) $Δ D B C ≅ Δ A C B$

(iv) $C M = \frac{1}{2} A B$

Q. In right triangle

A B C

, right angle is at

C, M

is the mid-point of hypotenuse

A B, C

is joined to

M

and produced to a point

D

such that

D M = C M

. Point

D

is joined to point

B

. Show that:

∠ D B C

is a right angle