Question

ABC is a right angled right triangle at C.a line through the midpoint M of hypotenuse AB and parallel to BC intersects AC at D.Show that
1)D is the mid point of AC
2)MD perpendicular AC
3)CM=MA=½AB.

Answer 1

GIVEN:-
ABC is a triangle right angled at C. A line passing through mid points M of hypotenuse AB and parallel to BC intersects AC at D

TO PROVE:-
(i) D is the midpoint of AC
(ii) CM=MA=1/2AB

PROOF:-
(i) M is a mid point of AB,
DM||CB

Therefore,
D is a midpoint of AC

(ii) Construction:- Join CM

In, ΔADM, ΔCDM

AD=DC => S
∠1= ∠3 => A
DM=DM [common] => S

∴ ΔADM ≅ ΔCDM [SAS ≅]
∴CM=AM=1/2AB [cpct]