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Quadrilaterals — Question 6

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Question 6

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 = 12AB\dfrac{1}{2}AB

Answer

(i) In Δ ABC,

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

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. NCERT Class 9 Mathematics CBSE Solutions.

By converse of mid-point theorem, we get :

Thus, D is the mid-point of AC.

Hence, proved that D is the mid-point of AC.

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

We know that,

Sum of co-interior angles = 180°.

⇒ ∠MDC + ∠DCB = 180°

⇒ ∠MDC + 90° = 180°

⇒ ∠MDC = 180° - 90°

⇒ ∠MDC = 90°

Hence, proved that MD ⊥ AC.

(iii) Join MC,

In Δ AMD and Δ CMD,

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

⇒ ∠ADM = ∠CDM (Since, MD ⊥ AC)

⇒ DM = DM (Common side)

∴ Δ AMD ≅ Δ CMD (By S.A.S. congruence rule)

We know that,

Corresponding parts of congruent triangles are equal.

⇒ AM = CM (By C.P.C.T.)

We know that,

M is the mid-point of AB.

∴ AM = 12AB\dfrac{1}{2}AB

⇒ CM = AM = 12AB\dfrac{1}{2}AB.

Hence, proved that CM = MA = 12AB\dfrac{1}{2}AB.

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Quadrilaterals — Interactive Study Guide

Parallelogram Properties

Opposite sides equal | Opposite angles equal | Diagonals bisect each other | Consecutive angles sum to 180°

Quick test: To check if a quadrilateral is a parallelogram, verify ANY ONE of these (or show one pair of opposite sides is both equal AND parallel).

Mid-Point Theorem

The line joining mid-points of two sides of a triangle is parallel to the third side and half its length.

If D, E are midpoints of AB, AC in ΔABC, then DE || BC and DE = ½BC.

Quick Self-Check

  1. Angle sum of a quadrilateral? (360°)
  2. ABCD is a parallelogram, ∠A = 75°. Find ∠B, ∠C, ∠D. (105°, 75°, 105°)
  3. In ΔPQR, M and N are midpoints of PQ and PR. QR = 12 cm. Find MN. (6 cm)

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