# 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 = 1/2 AB

**Solution:**

By converse of mid-point theorem, we know that a line drawn through the mid-point of any side of a triangle that is parallel to another side bisects the third side.

Let's construct the triangle according to the question.

**(i)** In ΔABC,

It is given that M is the mid-point of AB and MD || BC.

∴ D is the mid-point of AC. [Converse of mid-point theorem]

**(ii**) As DM || CB and AC is a transversal,

∠MDC + ∠DCB = 180° [Co-interior angles]

∠MDC + 90° = 180°

∠MDC = 90°

∴ MD ⊥ AC

**(iii)** Join MC

In ΔAMD and ΔCMD,

AD = CD (D is the mid-point of side AC)

∠ADM = ∠CDM (Each 90°)

DM = DM (Common)

∴ ΔAMD ≅ ΔCMD (By SAS congruence rule)

Therefore, AM = CM (By CPCT)

However, we also know that AM = 1/2 AB (M is the mid-point of AB)

∴ CM = AM = 1/2 AB

**☛ Check: **NCERT Solutions for Class 9 Maths Chapter 8

**Video Solution:**

## 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 = 1/2 AB

NCERT Maths Solutions Class 9 Chapter 8 Exercise 8.2 Question 7

**Summary:**

If ABC is a triangle right angled at C and a line through the mid-point M of hypotenuse AB and parallel to BC intersects AC at D, then D is the mid-point of AC, MD ⊥ AC, and CM = MA = 1/2 AB.

**☛ Related Questions:**

- ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rhombus.
- ABCD is a trapezium in which AB || DC, BD is a diagonal and E is the mid-point of AD. A line is drawn through E parallel to AB intersecting BC at F (see Fig. 8.30). Show that F is the mid-point of BC
- In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively (see the given figure). Show that the line segments AF and EC trisect the diagonal BD.
- Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.

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