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State true or false.

In $Δ A B C$ , $D$ is the midpoint of $B C$ . If $D L ⊥ A B$ and $D M ⊥ A C$ such that $D L = D M$ , then $A B = A C$ .

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True

False

answer is A.

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Detailed Solution

It is given that

D

is the midpoint of

B C

D L ⊥ A B

and

D M ⊥ A C

D L = D M

We have to show that,

A B = A C

.
As

D

is the midpoint of

B C

hence,

B D = C D

……(1)
Show that the triangles BDL and CMD are congruent.

∠ L = ∠ M

.…….(perpendicular hence

90 °

)

B D = C D

….….(by (1))

D L = D M

......(given)
Hence by RHL (right-angle-leg) postulate of congruency, these triangles are equal.
Thus,

Δ B D L ≅ Δ C M D

. ……(2)
Infer that since both the triangles are congruent hence all of its angles are equal.

∠ B = ∠ C

…….(3)
Now, if

2

angles of a triangle are equal.

⇒

Their corresponding sides are equal.
In

Δ A B C

A B = A C

…….(from (3))
Hence the statement is true.
Thereofore option 1 is correct.

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