AB = 8 cm, AC = 6 cm, AD = 7 cm, CD = 2.82 and CAB=50° (a) the length BC (b) The size of angle ABC; (c) The size of angle CAD (d) The area of triangle ACD
To solve this problem, we can use the Law of Cosines and the Law of Sines.
(a) To find the length BC, we can use the Law of Cosines. We have AB = 8 cm, AC = 6 cm, and the angle CAB = 50°.
Using the Law of Cosines, we have:
BC² = AB² + AC² - 2 * AB * AC * cos(CAB)
Substituting the given values, we get:
BC² = 8² + 6² - 2 * 8 * 6 * cos(50°)
Using a calculator, we can find the value of cos(50°) ≈ 0.6428:
BC² = 64 + 36 - 96 * 0.6428
BC² = 64 + 36 - 61.6608
BC² ≈ 38.3392
Taking the square root of both sides to find BC:
BC ≈ √38.3392
BC ≈ 6.2 cm
Therefore, the length BC is approximately 6.2 cm.
(b) To find the size of angle ABC, we can use the Law of Sines. We have AB = 8 cm, BC = 6.2 cm, and angle ABC = x°.
Using the Law of Sines, we have:
sin(ABC) / BC = sin(CAB) / AB
Substituting the given values, we get:
sin(x°) / 6.2 = sin(50°) / 8
Cross-multiplying, we have:
8 * sin(x°) = 6.2 * sin(50°)
Dividing both sides by 8:
sin(x°) = (6.2 * sin(50°)) / 8
Taking the inverse sine (sin⁻¹) on both sides:
x° ≈ sin⁻¹((6.2 * sin(50°)) / 8)
Using a calculator, we can find the approximate value of x°:
x° ≈ sin⁻¹((6.2 * sin(50°)) / 8)
x° ≈ sin⁻¹(0.7562)
x° ≈ 49.27°
Therefore, the size of angle ABC is approximately 49.27°.
(c) To find the size of angle CAD, we can use the Law of Sines. We have AC = 6 cm, AD = 7 cm, and angle CAD = y°.
Using the Law of Sines, we have:
sin(CAD) / AD = sin(CAB) / AC
Substituting the given values, we get:
sin(y°) / 7 = sin(50°) / 6
Cross-multiplying, we have:
6 * sin(y°) = 7 * sin(50°)
Dividing both sides by 6:
sin(y°) = (7 * sin(50°)) / 6
Taking the inverse sin (sin⁻¹) on both sides:
y° ≈ sin⁻¹((7 * sin(50°)) / 6)
Using a calculator, we can find the approximate value of y°:
y° ≈ sin⁻¹((7 * sin(50°)) / 6)
y° ≈ sin⁻¹(0.7453)
y° ≈ 47.63°
Therefore, the size of angle CAD is approximately 47.63°.
(d) To find the area of triangle ACD, we can use Heron's formula. We have AC = 6 cm, AD = 7 cm, and CD = 2.82 cm.
Heron's formula states:
Area = √(s * (s - AB) * (s - AC) * (s - AD))
Where s is the semi-perimeter of the triangle, which can be calculated as:
s = (AB + AC + AD) / 2
Substituting the given values, we get:
s = (8 + 6 + 7) / 2
s = 21 / 2
s = 10.5
Using Heron's formula, we have:
Area = √(10.5 * (10.5 - 8) * (10.5 - 6) * (10.5 - 7))
Calculating the expression within the square root:
Area = √(10.5 * 2.5 * 4.5 * 3.5)
Area = √(330.375)
Calculating the square root:
Area ≈ √330.375
Area ≈ 18.16 cm²
Therefore, the area of triangle ACD is approximately 18.16 cm².