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Assertion (A): In ΔABC, if ∠ABC = ∠DAC, AB = 8 cm, AC = 4 cm, AD = 5 cm, then BC = 3.5 cm.

Reason (R): SAS and ASS both are valid criteria for similarity of two triangles.

  1. A is true, R is false

  2. A is false, R is true

  3. Both A and R are true

  4. Both A and R are false

In ΔABC, if ∠ABC = ∠DAC, AB = 8 cm, AC = 4 cm, AD = 5 cm, then BC = 3.5 cm. Similarity of Triangles, RSA Mathematics Solutions ICSE Class 10.

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Answer

In ΔABC and ΔDAC,

∠ABC = ∠DAC (Given)

∠BCA = ∠DCA [Common angles]

ΔABC ∼ ΔDAC [By A.A. axiom]

Since, corresponding sides of similar triangle are proportional to each other.

⇒BAAD=BCAC⇒85=BC4⇒BC=85×4⇒BC=6.4 cm.\Rightarrow \dfrac{BA}{AD} = \dfrac{BC}{AC} \\[1em] \Rightarrow \dfrac{8}{5} = \dfrac{BC}{4} \\[1em] \Rightarrow BC = \dfrac{8}{5} \times 4 \\[1em] \Rightarrow BC = 6.4 \text{ cm}.

Assertion (A) is false.

SAS and ASS both are valid criteria for similarity of two triangles.

The SAS (Side-Angle-Side) criterion is a valid test for similarity.

The ASS (Angle-Side-Side) criterion is not a valid test for similarity.

Since the statement claims both are valid

Reason (R) is false.

Both A and R are false.

Hence, option 4 is the correct option.

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