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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

Name: 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 thati) D is the mid-point of AC ii) MD ⊥ AC iii) CM = MA = 1/2 AB
Uploaded: 2020-05-05T13:31:33Z
Duration: 5 min 37 s
Description: 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.

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.

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 thati) D is the mid-point of AC ii) MD ⊥ AC iii) CM = MA = 1/2 AB

(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:

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

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

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

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