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SAS Criteria for Congruency

ABC is a tria...

Question

ABC is a triangle right angled at C. A line through the mid point M of hypotenuse AB and parallel to BC intersect AC at D .Show that :
$C M = M A = \frac{1}{2} A B$

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Solution

$I n Δ A B C$

$M$ is the midpoint of $A B$

and $M D ∥ B C$
therefore $D$ is the midpoint of $A C$ [line drawn through midpoint of one side of a triangle parallel to another side, bisect the third side]

$\Rightarrow A D = D C ⟶ (1)$

also, $∠ A D M = ∠ C D M = 90^{o} (c o r r e s p o n d i n g a n g l e) I n Δ A D M a n d Δ C D M A D = C D (f r o m (1)) ∠ A D M = ∠ C D M = 90^{o} D M = D M (c o m m o n) ∴ Δ A D M ≅ Δ C D M (S A S) ∴ A M = C M (C P C T) ∵ A M = \frac{1}{2} A B (M i s t h e m i d p o i n t o f A B) ∴ C M = M A = \frac{1}{2} A B$

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