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Triangles — Question 8

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Question 8

In right triangle ABC, right angled at C, M is the mid-point of hypotenuse AB. C is joined to M and produced to a point D such that DM = CM. Point D is joined to point B. Show that :

(i) Δ AMC ≅ Δ BMD

(ii) ∠DBC is a right angle

(iii) Δ DBC ≅ Δ ACB

(iv) CM = 12AB\dfrac{1}{2}AB

In right triangle ABC, right angled at C, M is the mid-point of hypotenuse AB. C is joined to M and produced to a point D such that DM = CM. Point D is joined to point B. Show that : NCERT Class 9 Mathematics CBSE Solutions.
Answer

(i) In Δ AMC and Δ BMD,

⇒ AM = BM (M is the mid - point of AB)

⇒ ∠AMC = ∠BMD (Vertically opposite angles are equal)

⇒ CM = DM (Given)

∴ Δ AMC ≅ Δ BMD (By S.A.S. congruence rule)

Hence, proved that Δ AMC ≅ Δ BMD.

(ii) Since,

Δ AMC ≅ Δ BMD

∴ ∠ACM = ∠BDM (By C.P.C.T.)

From figure,

∠ACM and ∠BDM are alternate interior angles. Since alternate angles are equal, it can be said that DB || AC.

We know that,

Sum of co-interior angles = 180°.

⇒ ∠DBC + ∠ACB = 180°

⇒ ∠DBC + 90° = 180° [Since, ΔACB is right angled triangle at point C]

⇒ ∠DBC = 180° - 90°

∴ ∠DBC = 90°.

Hence, proved that ∠DBC is a right angle.

(iii) Since,

Δ AMC ≅ Δ BMD

∴ DB = AC (By C.P.C.T.)

In Δ DBC and Δ ACB,

⇒ DB = AC (Proved above)

⇒ ∠DBC = ∠ACB (Both equal to 90°)

⇒ BC = CB (Common)

∴ Δ DBC ≅ Δ ACB (By S.A.S. congruence rule)

Hence, proved that Δ DBC ≅ Δ ACB.

(iv) Since,

Δ DBC ≅ Δ ACB

⇒ AB = DC (By C.P.C.T.)

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

It is given that M is the midpoint of DC

⇒ CM = 12DC\dfrac{1}{2}DC = 12AB\dfrac{1}{2}AB

∴ CM = 12AB\dfrac{1}{2}AB

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

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Triangles — Interactive Study Guide

Congruence Criteria at a Glance

CriterionYou NeedRemember
SAS2 sides + included angleAngle MUST be between the 2 sides
ASA2 angles + included sideSide MUST be between the 2 angles
AAS2 angles + any sideSide can be anywhere
SSS3 sidesAll three sides match
RHSRight angle + hypotenuse + sideONLY for right triangles
NEVER use AAA (only proves similarity) or SSA (ambiguous case)!

Proof Writing Pattern

Every congruence proof follows this pattern:

  1. Identify the two triangles to compare.
  2. List three pairs of equal elements (sides/angles) with reasons.
  3. State the criterion (SAS/ASA/AAS/SSS/RHS).
  4. Conclude congruence.
  5. Use CPCT for any further deduction.

Quick Self-Check

  1. In ΔABC, AB = AC. ∠B = 55°. Find ∠A. (∠C = 55°, ∠A = 70°)
  2. Can a triangle have sides 2, 3, 6? (No: 2+3=5 < 6, violates triangle inequality)
  3. In a triangle, the longest side is opposite to which angle? (The largest angle)

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