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:
$△ A M C ≅ △ B M D$

Question

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:
$△ A M C ≅ △ B M D$

Answer 1

In right angled $△ A B C$ ,
$∠ C = 9 0^{\circ}$ ,

$M$ is the mid-point of $A B$ i.e, $A M = M B$ & $D M = C M$
In $△ A M C$ and $△ B M D$ ,
$A M = B M$ ------ (M is the mid-point)
$∠ C M A = ∠ D M B$ ------ (Vertically opposite angles)
$C M = D M$ ------ (Given)

$∴ △ A M C ≅ △ B M D$ , by SAS Postulate

Answer 2

In right angled

△ A B C

,

∠ C = 9 0^{\circ}

,

M

is the mid-point of

A B

i.e,

A M = M B

&

D M = C M

In

△ A M C

and

△ B M D

,

A M = B M

------ (M is the mid-point)

∠ C M A = ∠ D M B

------ (Vertically opposite angles)

C M = D M

------ (Given)

∴ △ A M C ≅ △ B M D

, by SAS Postulate

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:
$△ A M C ≅ △ B M D$

Answer

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In right angled △ABC, ∠C=90∘, M is the mid-point of AB i.e, AM=MB & DM=CMIn △AMC and △BMD,AM=BM ------ (M is the mid-point)∠CMA=∠DMB ------ (Vertically opposite angles)CM=DM ------ (Given) ∴△AMC≅△BMD, by SAS Postulate

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