Question

in right triangle ABC, right-angled at C, M is the midpoint 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. show that:
1) ΔAMC ≅ ΔBMD
2) ∠DBC is a right angle
3) Δ DBC ≅ ΔACB
4) CM = 1/2 AB

Answer 1

Step-by-step explanation:

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

DC ² =DB²+BC ² (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= 1 /2 AB

As △DBC≅△ACB

CM= DC/2

∴DC=AB(△DBC≅△ACB)

So, CM= AB/2

∴CM= 1/2 AB