In a triangle ABC, BC = 8 cm and AC = 12 cm and angle ABC = 120.
a. Calculate the length of AB, correct to one decimal place.
b. If BC is the base of the triangle, calculate, correct to one decimal place;
(i) the perpendicular height of triangle;
(ii) the area of the triangle;
(iii) the size of angle ACB.
a) use the sine law to find angle A
sinA/8 = sin 120°/12
sinA = .57735..
angle A = appr 35.3° and angle C = appr 24.7° <----- b iii)
b i) make your sketch and see that
perp. height/12 = sin C
perp.height = 12 sin 24.7 = appr 5.02
ii) area of triangle = (1/2)(12)(8)sin24.7° = ....
iii) see above
The given data is often called the ambiguous case and often results in two different
answers if the sine law is use. However since we can have only one obtuse angle in a triangle
and one angle was 120°, we don't have to worry about that situation.
a. To find the length of AB, we can use the Law of Cosines. According to the Law of Cosines, in a triangle with sides a, b, and c, and angle C opposite side c, we have the formula:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, we can label AB as side c, AC as side a, and BC as side b. Also, the angle opposite side c (AB) is angle ABC, which is 120 degrees. Plugging in the values, we can calculate the length of AB:
AB^2 = AC^2 + BC^2 - 2 * AC * BC * cos(ABC)
AB^2 = 12^2 + 8^2 - 2 * 12 * 8 * cos(120)
AB^2 = 144 + 64 - 192 * (-0.5)
AB^2 ≈ 144 + 64 + 96
AB^2 ≈ 304
AB ≈ √304
AB ≈ 17.4 cm (rounded to one decimal place)
So, the length of AB is approximately 17.4 cm.
b. (i) To find the perpendicular height (h) of the triangle, we can choose any base and draw the perpendicular from the opposite vertex. Since BC is given as the base, we will find the height from B to AC.
To find the height, we can use the formula for the area of a triangle:
Area = (1/2) * base * height
Area = (1/2) * BC * h
h = (2 * Area) / BC
We need to calculate the area first to find the height.
(ii) To find the area of the triangle, we can use Heron's Formula, which states that the area of a triangle with sides a, b, and c is given by:
Area = sqrt(s(s-a)(s-b)(s-c))
where s is the semiperimeter of the triangle:
s = (a + b + c) / 2
In this case, we have sides a = AC = 12cm, b = BC = 8cm, and c = AB ≈ 17.4cm. Calculating the values:
s = (12 + 8 + 17.4) / 2
s ≈ 37.4 / 2
s ≈ 18.7
Now we can find the area:
Area = sqrt(18.7(18.7-12)(18.7-8)(18.7-17.4))
Area ≈ sqrt(18.7(6.7)(10.7)(1.3))
Area ≈ sqrt(1567.1971)
Area ≈ 39.6 cm^2 (rounded to one decimal place)
Now, let's calculate the height:
h = (2 * Area) / BC
h = (2 * 39.6) / 8
h ≈ 4.95 cm (rounded to one decimal place)
So, (i) the perpendicular height of the triangle is approximately 4.95 cm.
(iii) To find the size of angle ACB, we can use the Law of Cosines again. The formula for the Law of Cosines is:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, we can label AB as side c, AC as side a, and BC as side b. Also, the angle opposite side c (AB) is angle ABC, which is 120 degrees. Plugging in the values, we can calculate the size of angle ACB:
cos(C) = (a^2 + b^2 - c^2) / (2ab)
cos(ACB) = (12^2 + 8^2 - 17.4^2) / (2 * 12 * 8)
cos(ACB) = (144 + 64 - 302.76) / 192
cos(ACB) ≈ -0.1116
ACB ≈ cos^(-1)(-0.1116)
ACB ≈ 101.9 degrees (rounded to one decimal place)
So, (iii) the size of angle ACB is approximately 101.9 degrees.
To calculate the length of AB in triangle ABC, you can use the Cosine Rule. The Cosine Rule states that in any triangle ABC with sides a, b, and c and angle C opposite side c, the following formula can be used:
c^2 = a^2 + b^2 - 2ab * cos(C)
a. Calculate the length of AB, correct to one decimal place:
Given:
BC = 8 cm
AC = 12 cm
∠ABC = 120°
Let's substitute the given values into the Cosine Rule formula:
AB^2 = BC^2 + AC^2 - 2 * BC * AC * cos(∠ABC)
AB^2 = 8^2 + 12^2 - 2 * 8 * 12 * cos(120°)
AB^2 = 64 + 144 - 192 * (-0.5)
AB^2 = 64 + 144 + 96
AB^2 = 304
Now, take the square root of both sides to find AB:
AB = √304
AB ≈ 17.5 cm (rounded to one decimal place)
b. If BC is the base of the triangle, calculate, correct to one decimal place:
(i) To find the perpendicular height of the triangle, you can use the formula for the area of a triangle:
Area = 0.5 * base * height
In this case, the base is BC and the area is given as:
Area = 0.5 * BC * height
To find the height, we need to rearrange the formula:
height = (2 * Area) / BC
(ii) To find the area of the triangle, we can use Heron's formula:
Area = √(s * (s - AB) * (s - BC) * (s - AC))
where s is the semi-perimeter of the triangle, calculated as:
s = (AB + BC + AC) / 2
(iii) To find the size of angle ACB, we can use the Sine Rule. The Sine Rule states that in any triangle ABC with sides a, b, and c and angles A, B, and C, the following formula can be used:
sin(A) / a = sin(B) / b = sin(C) / c
In this case, we can rearrange the formula to find angle ACB:
sin(∠ACB) = (BC * sin(∠ABC)) / AB
ACB = arcsin((BC * sin(∠ABC)) / AB)