Question

In rt. triangle ABC , right angled at B, AB -5cm and
angle ACB - 30°. Determine the length of the sides BC and AC. ​

Answer 1

Answer:

The lengths of the sides BC & AC are 5 √3 cm & 10 cm respectively.

Step-by-step-explanation:

NOTE: Refer to the attachment for the diagram.

In figure, △ABC is a right-angled triangle.

m∠ABC = 90°

m∠ACB = 30°

AB = 5 cm

We have to find the lengths of the sides BC & AC.

Now, in △ABC,

m∠A + m∠B + m∠C = 180° - - [ Angle sum property of triangle ]

⇒ m∠A + 90° + 30° = 180°

⇒ m∠A + 120° = 180°

⇒ m∠A = 180° - 120°

⇒ m∠A = 60°

∴ △ABC is a 30°-60°-90° triangle.

Now, by 30°-60°-90° triangle theorem,

AB = ½ × AC - - [ Side opposite to 30° ]

⇒ 5 = ½ × AC

⇒ AC = 5 × 2

⇒ AC = 10 cm

Now,

BC = ( √3 / 2 ) × AC - - [ Side opposite to 60° ]

⇒ BC = √3 / 2 × 10

⇒ BC = √3 × 10 ÷ 2

⇒ BC = √3 × 5

⇒ BC = 5 √3 cm

∴ The lengths of the sides BC & AC are 5 √3 cm & 10 cm respectively.

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Alternative Method:

Let ∠ACB = θ

∴ θ = 30°

Now, from the trigonometric table, we know that,

sin 30° = 1 / 2 - - ( 1 )

cos 30° = √3 / 2 - - ( 2 )

Now, in △ABC,

sin θ = Opposite side / Hypotenuse

⇒ sin θ = AB / AC

⇒ sin 30° = 5 / AC

⇒ 1 / 2 = 5 / AC - - [ From ( 1 ) ]

⇒ AC = 5 × 2

⇒ AC = 10 cm

Now, in △ABC,

cos θ = Adjacent side / Hypotenuse

⇒ cos θ = BC / AC

⇒ cos 30° = BC / 10

⇒ √3 / 2 = BC / 10 - - [ From ( 2 ) ]

⇒ √3 / 2 × 10 = BC

⇒ BC = √3 × 10 ÷ 2

⇒ BC = √3 × 5

⇒ BC = 5 √3 cm

∴ The lengths of the sides BC & AC are 5 √3 cm & 10 cm respectively.