Question

ABC is a right angled triangle in which B=90 degrees and C=2A. If BC= 12 cm then ar(ABC) is

Answer 1

ABC is a right angled triangle in which angle B = 90° and angle C =2 × angle A

we know,
sum of all angles of any triangle = 180°

so, angle A + angle B + angle C = 180°

angle A + 90° + 2 × angle A = 180° [ from question, C = 2A and B = 90°]

3 × angle A + 90° = 180°

3 × angle A = 90°

angle A = 30°

and angle C = 2 × angle A = 60°

now, use sine formula ,

e.g., $\frac{sinA}{a}=\frac{sinB}{b}=\frac{sinC}{c}$

here, BC = a , AB = c , and CA = b

so, sin30°/BC = sin60°/AB

(1/2)/12 = (√3/2)/AB

1/24 = √3/2AB

AB = 12√3 cm

hence, area of ∆ABC = 1/2 × BC × AB

= 1/2 × 12 × 12√3

= 72√3 cm²

Answer 2

ABC is a right angled triangle in which angle B = 90° and angle C =2 × angle A

we know,

sum of all angles of any triangle = 180°

so, angle A + angle B + angle C = 180°

angle A + 90° + 2 × angle A = 180° [ from question, C = 2A and B = 90°]

3 × angle A + 90° = 180°

3 × angle A = 90°

angle A = 30°

and angle C = 2 × angle A = 60°

now, use sine formula ,

e.g.,

here, BC = a , AB = c , and CA = b

so, sin30°/BC = sin60°/AB

(1/2)/12 = (√3/2)/AB

1/24 = √3/2AB

AB = 12√3 cm

hence, area of ∆ABC = 1/2 × BC × AB

= 1/2 × 12 × 12√3

= 72√3 cm²