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Standard IX

Mathematics

The Mid-Point Theorem

In Δ ABC, D i...

Question

$I n Δ ABC, D is the midpoint of BC. if D L ⊥ A B a n d D M ⊥ A C such that DL = DM, prove that AB = AC.$

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Solution

ANSWER:
In $Δ B D L a n d Δ$ CDM, we have:BD=CD (D is midpoint)DL=DM
⇒BL=MC (CPCT)
∴ $∠ B = ∠ C$
This makes triangle ABC an isosceles triangle.
Or AB = AC
Hence, proved.

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Similar questions

In Figure given below, $A B C D$ is a parallelogram, $D L ⊥ A B$ and $D M ⊥ B C .$ If $A B = 18 c m,$ $B C = 12 c m$ and $D M = 9.3 c m,$ find $D L .$

Q. In the given fig. ABCD is a parallelogram.

D L ⊥ A B

and

D M ⊥ B C

If AB = 18 cm, BC = 12 cm and DM = 10 cm. Find DL.

Q. (a)
In Fig.,

D

is a point on hypotenuse

A C

Δ A B C,

such that

B D ⊥ A C, D M ⊥ B C and D N ⊥ A B .

Prove that :

(i) D M^{2} = D N . M C

(b)
In Fig.,

D

is a point on hypotentenuse

A C

Δ A B C,

such that

B D ⊥ A C, D M ⊥ B C and D N ⊥ A B . Prove that :

(ii)

D N^{2} = D M \times A N

Q. In Fig.,

D

is a point on hypotenuse

A C

△ A B C

, such that

B D ⊥ A C, D M ⊥ B C

and

D N ⊥ A B .

Prove that :
(i)

D M^{2} = D N . M C

(ii)

D N^{2} = D M . A N

Q. In figure,

D

is a point on hypotenuse

A C

△ A B C

, such that

B D ⊥ A C

D M ⊥ B C

D N ⊥ A B

. Prove that:
(i)

{D M}^{2} = D N . M C

(ii)

{D N}^{2} = D M . A N

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In Δ ABC, D is the midpoint of BC. if DL⊥ AB and DM⊥AC such that DL = DM, prove that AB = AC.

ANSWER: In ΔBDLandΔCDM, we have:BD=CD (D is midpoint)DL=DM ⇒BL=MC (CPCT) ∴ ∠B=∠C This makes triangle ABC an isosceles triangle. Or AB = AC Hence, proved.

$I n Δ ABC, D is the midpoint of BC. if D L ⊥ A B a n d D M ⊥ A C such that DL = DM, prove that AB = AC.$

ANSWER:
In $Δ B D L a n d Δ$ CDM, we have:BD=CD (D is midpoint)DL=DM
⇒BL=MC (CPCT)
∴ $∠ B = ∠ C$
This makes triangle ABC an isosceles triangle.
Or AB = AC
Hence, proved.