Question

"Question 7 ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallel to BC intersects AC at D. Show that (i) D is the mid-point of AC (ii) MD ⊥ AC (iii) CM = MA = AB/2

Class 9 - Math - Quadrilaterals Page 151"

Answer 1

Converse of mid point theorem:

The line drawn through the midpoint of one side of a triangle, parallel to another side bisect the third side.

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Given;

ABC is a right angled triangle in which ∠C = 90° and M is the midpoint of AB.

Also a line through the midpoint M of hypotenuse AB and parallel to BC intersects AC at D such that MD||BC

To show:

 (i) D is the mid-point of AC

 (ii) MD ⊥ AC

(iii) CM = MA = AB/2

Proof:

(i)  In ΔACB,

M is the mid point of AB and MD || BC

So, D is the mid point of AC.

(by Converse of mid point theorem)

 

(ii)  Given,MD||BC & CD is a

transversal.

∠ACB = ∠ADM (Corresponding angles)

∠ACB = 90°(given)

∴ ∠ADM = 90°  

⇒ MD ⊥ AC

(iii) In ΔAMD and ΔCMD,

(M is mid point of AB)

From eq i & ii,

CM = AM =  1/2 AB

Hope this will help you...