ABC is a triangle right-angled at C. A line through the mid-point M of hypotenuse AB parallel to BC intersects AC … AC (iii) CM =…
Answers:
Given: A △ABC , right – angled at C. A line through the mid – point M of hypotenuse AB parallel to BC intersects AC at D.
To Prove:
(i) D is the mid – point of AC
(ii) MD | AC
(iii) CM = MA = 1 / 2 AB.
Proof : (i) Since M is the mid point of hyp. AB and MD | | BC .
⇒ D is the mid – point of AC .
(ii) Since ∠BCA = 90°
and MD | | BC [given]
⇒ ∠MDA = ∠BCA
= 90° [corresp ∠s]
⇒ MD | AC
(iii) Now, in △ADM and △CDM
MD = MD [common]
∠MDA = ∠MDC [each = 90°]
AD = CD [∵ D the mid – point of AC]
⇒ △ADM ≅ △CDM [by SAS congruence axiom]
⇒ AM = CM
Also, M is the mid – point of AB [given]
⇒ CM = MA = 1 / 2 = AB.