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