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Byju's Answer

Standard IX

Mathematics

The Mid-Point Theorem

A triangle ...

Question

A triangle $A B C$ in which $A B = A C,$ $M$ is a point on $A B$ and $N$ is a point on $A C$ such that if $B M = C N$ then $A M = A N .$

True

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False

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Solution

The correct option is A True
Since $M$ is the midpoint of $A B$ and $N$ is the midpoint of $A C,$ we have
$A B = A M + M B$ and $A B = A N + N C$

So,
$\begin{matrix} A B & = A C [Given] A M + B M & = A N + C M A M + B M & = A N + B M [Given B M = C N] A M & = A N \end{matrix}$

Hence proved, $A M = A N .$

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Similar questions

The given figure shows a triangle ABC in which AB = AC. M is a point on AB and N is a point on AC such that BM = CN.

Hence,AM = AN

State true or false.

Q. The following figure shows a triangle

A B C

in which

A B = A C

M

is a point on

A B

and

N

is a point on

A C

such that

B M = C N

. Prove that :
i)

A M = A N

ii)

Δ A M C ≅ Δ A N B

iii)

B N = C M

iv)

Δ B M C ≅ Δ C N B

The given figure shows a triangle $A B C$ in which $A B = A C$ . $M$ is a point on $A B$ and $N$ is a point on $A C$ such that $B M = C N$ . Hence, $B N = C M$ .

If the above statement is true, then mention answer as 1, else mention 0 if false.

Q. In △ABC, M and N are points on AB and AC respectively such that BM = CN. If ∠B = ∠C then show that MN∥BC.

In $Δ A B C$ ; $B M ⊥ A C$ and $C N ⊥ A B$ ;

Then, $\frac{A B}{A C} = \frac{B M}{C N} = \frac{A M}{A N}$

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A triangle ABC in which AB=AC, M is a point on AB and N is a point on AC such that if BM=CN then AM=AN.

The correct option is A TrueSince M is the midpoint of AB and N is the midpoint of AC, we haveAB=AM+MB and AB=AN+NCSo,AB=AC[Given]AM+BM=AN+CMAM+BM=AN+BM[Given BM=CN]AM=ANHence proved, AM=AN.

A triangle $A B C$ in which $A B = A C,$ $M$ is a point on $A B$ and $N$ is a point on $A C$ such that if $B M = C N$ then $A M = A N .$