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: [tex]\frac{sinA}{BC}[/tex] = [tex]\frac{sinB}{AC}[/tex] = [tex]\frac{sinC}{AB}[/tex]
∵ AB = 9 units , m∠A = 60° , m∠B = 75°, m∠C = 45°
∴ [tex]\frac{sin(60)}{BC}[/tex] = [tex]\frac{sin(75)}{AC}[/tex] = [tex]\frac{sin(45)}{9}[/tex]
- 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 = [tex]\frac{1}{2}[/tex] (AB)(BC) sin B
∴ Area of the triangle ABC = [tex]\frac{1}{2}[/tex] (9)(11) sin(75)
∴ Area of the triangle ABC = 47.81 ≅ 48 unit²
* Area of the triangle ABC is 48 units²
