Question

In triangle ABC, AC=12cm, Angle ABC=90, Angle BAC=30. Then find AB & BC.

Answer 1

Answer:

AB =  6√3cm

BC = 6cm

Step-by-step explanation:

Given,

Δ ABC, with AC = 12cm,  ∠ABC = 90° and ∠BAC = 30°

To find,

The length of the sides AB and BC

Solution:

Recall the formula

sin θ = $\frac{Opposite \ side }{hypotenuse}$

cost θ = $\frac{adjacent \ side }{hypotenuse}$

sin θ = $\frac{1}{2}$

cos θ = $\frac{\sqrt{3} }{2}$

Since Δ ABC is right angled triangle right angled at B, the hypotenuse of the triangle = AC = 12cm

and by the trigonometric identities, we get

sin 30 = $\frac{Opposite \ side }{hypotenuse}$ = $\frac{BC}{AC}$

Substituting the value of sin 30 and the value of AC, we get

$\frac{1}{2}$ = $\frac{BC}{12}$

BC = $\frac{12}{2}$ = 6

The length of the side BC =  6cm

Again we have,

cost 30 = $\frac{adjacent \ side }{hypotenuse}$ = $\frac{AB}{AC}$

Substituting the value of cos 30 and the value of AC, we get

$\frac{\sqrt{3} }{2}$ = $\frac{AB}{12}$

2AB = 12√3

AB = 6√3

The length of the side AB =  6√3cm

∴ AB =  6√3cm

BC = 6cm

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Answer 2

Answer:

$AB = 6\sqrt{3} cm \\BC = 6 cm$

Step-by-step explanation:

Given :-

• ΔABC , l(AC) = 12 cm ,
• ∠ABC = 90° , ∠BAC = 30°

To find :-

l(AB) , l(BC)

Solution :-

1. From fig. we can see that l(AC) = 12 cm , is hypotenuse  .

2. By using trigonometric identities ,

$sin(60) = \frac{AB}{12} \\AB = 12 * sin(60) \\AB = 6\sqrt{3}$ ---- (1)

3. $cos(60) = \frac{BC}{12} \\BC = 12* cos (60) \\BC= 6 cm$ ---- (2)

4. Therefore , answer is

$AB = 6\sqrt{3} cm \\BC = 6 cm$

#SPJ3