Question

Given: △ABC, m∠A=60°,

Answer 1

Answer:

Perimeter of ΔABC is 32 units

Area of the triangle ABC is 48 units²

Step-by-step explanation:

- In triangle ABC

∵ m∠A = 60°

∵ m∠C = 45°

- The sum of the measures of the interior angles of any Δ is 180°

∴ m∠A + m∠B + m∠C = 180°

- Substitute the measure of angles A and C to find the measure

  of angle B

∴ 60 + m∠B + 45 = 180

∴ m∠B + 105 = 180

- Subtract 105 from both sides

∴ m∠B = 75°

- We need to find the perimeter and the area of the ΔABC

- Let us use the sine rule to find the lengths of sides BC and AC

→ sine rule: $\frac{sinA}{BC}$ = $\frac{sinB}{AC}$ = $\frac{sinC}{AB}$

∵ AB = 9 units , m∠A = 60° , m∠B = 75°, m∠C = 45°

∴ $\frac{sin(60)}{BC}$ = $\frac{sin(75)}{AC}$ = $\frac{sin(45)}{9}$

- By using cross multiplication

∴ BC × sin(45) = 9 × sin(60)

- Divide both sides by sin(45)

∴ BC = 11.02 ≅ 11 units

∴ AC × sin(45) = 9 × sin(75)

- Divide both sides by sin(45)

∴ AC = 12.29 ≅ 12 units

- The perimeter of the triangle is the sum of the lengths of its 3 sides

∴ Perimeter of ΔABC = AB + BC + AC

∴ Perimeter of ΔABC = 9 + 11 + 12

∴ Perimeter of ΔABC = 32 units

* Perimeter of ΔABC is 32 units

- Let us find the area by using the sine rule

∵ Area of the triangle ABC = $\frac{1}{2}$ (AB)(BC) sin B

∴ Area of the triangle ABC = $\frac{1}{2}$ (9)(11) sin(75)

∴ Area of the triangle ABC = 47.81 ≅ 48 unit²

* Area of the triangle ABC is 48 units²