In right triangle ABC, right angled at C, M is the mid-point of
hypotenuse AB. C is joined to M and produced to a point D
such that DM = CM. Point D is joined to point B (see the given figure).Show that:
(i)
AMC =BMD
(ii)
<DBC is a right angle.
(iii)
DBC =ABC
(iv) CM =1/2 AB
Share
i) △AMC≅△BMD
Proof: As 'M' is the midpoint
BM=AM
And also it is the mid point of DC then
DM=MC
And AC=DB (same length)
∴Therefore we can say that
∴△AMC≅△BMD
ii) ∠DBC is a right angle
As △DBC is a right angle triangle and
DC2=DB2+BC2 (Pythagoras)
So, ∠B=90°
∴∠DBC is 90°
iii) △DBC≅△ACB
As M is the midpoint of AB and DC. So, DM=MC and AB=BM
∴DC=AB (As they are in same length)
And also, AC=DB
and ∠B=∠C=90°
By SAS Axiom
∴△DBC≅△ACB
iv) CM=21AB
As △DBC≅△ACB
CM=2DC
∴DC=AB(△DBC≅△ACB)
So, CM=2AB
∴CM=21AB
Answer: