Question

13. The diagram shows the support bracket for a restaurant sign. AB=60 cm, AC=109 cm and ZBAD=41°. A 41° 109 cm 60 cm NOT TO SCALE B С D THE BROTHERS CONCH DINNERS Calculate (a) the length of BC [3] [3] (b) the angle C (c) [3] the length of AD​

Answer 1

Answer:

(a) BC = 91 cm

(b) ∠C = 33.4° (nearest tenth)

(c) AD = 79.5 cm (nearest tenth)

Step-by-step explanation:

(a) Pythagoras' Theorem:  a² + b² = c²

(where a and b are the legs, and c is the hypotenuse, of a right triangle)

Given:

• a = AB = 60 cm
• b = BC
• c = AC = 109 cm

⇒ 60² + BC² = 109²

⇒ 3600 + BC² = 11881

⇒ BC² = 11881 - 3600

⇒ BC² = 8281

⇒ BC = √(8281)

⇒ BC = 91 cm

(b) Sine rule to find an angle:

   $\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}$

(where A, B and C are the angles, and a, b and c are the sides opposite the angles)

Given:

• ∠B = 90°
• b = AC = 109 cm
• c = AB = 60 cm

$\implies \dfrac{\sin (90)}{109}=\dfrac{\sin C}{60}$

$\implies \sin C=60 \cdot\dfrac{\sin (90)}{109}$

$\implies \sin C=\dfrac{60}{109}$

$\implies C=33.39848847...\textdegree$

$\implies C=33.4\textdegree \ \sf(nearest \ tenth)$

(c) Sine rule to find a side length:

   $\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}$

(where A, B and C are the angles, and a, b and c are the sides opposite the angles)

Sum of interior angles of a triangle = 180°

Given:

• ∠B = 90°
• b = AD
• ∠D= 180° - 41° - 90° = 49°
• d = AB = 60 cm

$\implies \dfrac{AD}{\sin (90)}=\dfrac{60}{\sin (49)}$

$\implies AD=\sin (90) \cdot \dfrac{60}{\sin (49)}$

$\implies AD=79.5007796...$

$\implies AD=79.5 \ \sf cm \ (nearest \ tenth)$