Question

Assertion(A): In right triangles ABC and DEF (∠C = ∠F = 90°), ∠B and ∠E are acute angles, such that sin B = sin E, then ∠B = ∠E Reason (R): ΔABC ~ ΔDEF​

Answer 1

To Find :-

(A) both A & R are true and R is the correct explanation of A.

(B) both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true .

Solution :-

In ∆ABC,

→ sin B = Perpendicular / Hypotenuse

→ sin B = AC/AB

In ∆DEF,

→ sin E = Perpendicular / Hypotenuse

→ sin E = DF/DE

since,

→ sin B = sin E (given)

→ AC/AB = DF/DE

→ AC/DF = AB/DE = Let k ---------- Eqn.(1)

→ AC = k•DF and AB = k•DE ------- Eqn.(2)

now, in ∆ABC,

→ AB² = AC² + BC² { By pythagoras theorem .}

→ BC = √(AB² - AC²)

putting value from Eqn.(2) ,

→ BC = √[(k•DE)² - (k•DF)²]

→ BC = √[k²(DE² - DF²)

→ BC = k√(DE² - DF²) -------- Eqn.(3)

in ∆DEF,

→ DE = DF² + EF² { By pythagoras theorem .}

→ EF = √(DE² - EF²) -------- Eqn.(4)

dividing Eqn.(3) by Eqn.(4) ,

→ BC/EF = [k√(DE² - DF²)]/√(DE² - DF²)

→ BC/EF = k -------- Eqn.(5)

from Eqn.(1) and Eqn.(5) we get,

→ AC/DF = AB/DE = BC/EF = k

→ AC/DF = AB/DE = BC/EF

as we can see that, corresponding sides of both ∆'s are in same ratio .

therefore, we can conclude that,

→ ∆ABC ~ ∆DEF { By SSS similarity. }

hence,

→ ∠B = ∠E { When two ∆'s are similar, their corresponding angles are congruent . }

given that,

→ Assertion (A) :- In right triangles ABC and DEF (∠C = ∠F = 90°), ∠B and ∠E are acute angles, such that sin B = sin E, then ∠B = ∠E .

• True . { Proved above. }

Reason (R) :- ΔABC ~ ΔDEF .

• True . { Proved above. }

∴ (A) Both A & R are true and R is the correct explanation of A.

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