AB = 8 cm, AC = 6 cm, AD = 7 cm, CD = 2.82 and CAB=50°.
Calculate to 2 decimal places
(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. Let's go step by step to calculate each value.
(a) The length BC:
Using the Law of Cosines, we can find the length BC by applying the formula:
BC^2 = AB^2 + AC^2 - 2 * AB * AC * cos(CAB)
Substituting the given values, we get:
BC^2 = (8 cm)^2 + (6 cm)^2 - 2 * 8 cm * 6 cm * cos(50°)
Now we can calculate BC:
BC^2 = 64 cm^2 + 36 cm^2 - 96 cm^2 * cos(50°)
BC^2 = 100 cm^2 - (96 cm^2 * 0.6428)
BC^2 ≈ 38.86 cm^2
Taking the square root of both sides:
BC ≈ √(38.86 cm^2)
BC ≈ 6.23 cm
So, the length BC is approximately 6.23 cm.
(b) The size of angle ABC:
Using the Law of Sines, we have:
sin(CAB) / BC = sin(ABC) / AC
Substituting the known values:
sin(50°) / 6.23 cm = sin(ABC) / 6 cm
Now we can solve for sin(ABC):
sin(ABC) = (sin(50°) / 6.23 cm) * 6 cm
sin(ABC) ≈ 0.5706
To find the angle ABC, we need to take the inverse sine (sin^-1) of sin(ABC):
ABC = sin^-1(0.5706)
ABC ≈ 34.54°
Therefore, the size of angle ABC is approximately 34.54°.
(c) The size of angle CAD:
Since angle CAD is opposite side CD, we can use the Law of Sines again:
sin(CAD) / CD = sin(ACD) / AD
Substituting the given values:
sin(CAD) / 2.82 cm = sin(ACD) / 7 cm
Now we can solve for sin(CAD):
sin(CAD) = (sin(ACD) / 7 cm) * 2.82 cm
sin(CAD) ≈ 0.4026
Taking the inverse sine:
CAD = sin^-1(0.4026)
CAD ≈ 23.52°
Therefore, the size of angle CAD is approximately 23.52°.
(d) The area of triangle ACD:
To calculate the area of triangle ACD, we can use Heron's Formula, which requires the lengths of all three sides:
s = (AB + AC + BC) / 2
s = (8 cm + 6 cm + 6.23 cm) / 2
s ≈ 10.11 cm (rounded to two decimal places)
Now we can calculate the area:
Area of ACD = √[s(s - AB)(s - AC)(s - CD)]
Area of ACD = √[10.11 cm(10.11 cm - 8 cm)(10.11 cm - 6 cm)(10.11 cm - 2.82 cm)]
Area of ACD ≈ √[10.11 cm * 2.11 cm * 4.11 cm * 7.29 cm]
Area of ACD ≈ √[324.67 cm^4]
Area of ACD ≈ 18.02 cm^2
Therefore, the area of triangle ACD is approximately 18.02 cm^2.