Question

PLEASE WILL GIVE 50 POINTS PLEASE ANSWER ​

Answer 1

AB = 75

BC = 60

AC = 45

m∠A = 53°

m∠B = 37°

m∠C = 90°

Explanation:

Trigonometric ratios

`\sf \sin(\theta)=(O)/(H)\quad\cos(\theta)=(A)/(H)\quad\tan(\theta)=(O)/(A)`

where:

• 
  `\theta` is the angle
• O is the side opposite the angle
• A is the side adjacent the angle
• H is the hypotenuse (the side opposite the right angle)

Given:

`\sf \tan(A)=(60)/(45)`

Therefore:

• side opposite angle A = BC = 60
• side adjacent angle A = AC = 45

To find the length of AB (the hypotenuse), use Pythagoras’ Theorem:

`a^2+b^2=c^2`

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

⇒ AC² + BC² = AB²

⇒ 45² + 60² = AB²

⇒ AB² = 5625

⇒ AB = √5625

⇒ AB = 75

To find m∠A:

`\implies\sf \tan(A)=(60)/(45)`

`\implies\sf A=\tan^(-1)\left((60)/(45)\right)`

`\implies\sf A=53^(\circ)\:(nearest\:degree)`

m∠C = 90° (as it is a right angle)

The interior angles of a triangle sum to 180°

⇒ m∠A + m∠B + m∠C = 180°

⇒ 53° + m∠B + 90° = 180°

⇒ m∠B = 180° - 53° - 90°

⇒ m∠B = 37°