Question

The airline distance from Rome, Italy to Paris, France is 697 miles. The distance from Paris to Berlin,

Germany is 545 miles and the distance from Berlin to Rome is 734 miles. What is the angle formed from

Berlin to Paris to Rome?

Answer 1

• The Angle formed from Rome to Berlin(R)=44.68 degrees
• The Angle formed from Paris to Berlin(P)=71.26 degrees
• The Angle formed from Berlin to Paris(B)=64.06 degrees

Explanation:

The movement of the airplane can be represented using a triangle. In Triangle PRB, we are required to find Angles P, R and B.

Using Law of Cosines

`cos R=(p^2+b^2-r^2)/(2pb) \\cos R=(734^2+697^2-545^2)/(2*734*697)\\R=arcCos\left((734^2+697^2-545^2)/(2*734*697)\right)\\R=44.68^0`

Using law of Sines

`(r)/(sin R)=(p)/(SIn P) \\(545)/(sin 44.68)=(734)/(Sin P) \\\text{Cross Multiply}\\545 X Sin P=734 X sin 44.68\\Sin P=(734 X sin 44.68)/(545) \\P=arcSin\left((734 X sin 44.68)/(545)\right)\\P=71.26^0`

Now,

P+R+B=180 degrees (Sum of Angles in a Triangle)

`71.26^0+44.68^0+B=180^0\\B=180^0-[71.26^0+44.68^0]\\B=64.06^0`

• The Angle formed from Rome to Berlin(R)=44.68 degrees
• The Angle formed from Paris to Berlin(P)=71.26 degrees
• The Angle formed from Berlin to Paris(B)=64.06 degrees

Answer 2

Angle Rome to Paris = 44.68°

Angle Paris to Berlin = 71.26°

Angle Berlin to Rome = 64.06

Explanation:

See attachment for diagram illustration for this.

Let represent distance from Rome to Paris as a, distance from Paris to Berlin as b, and distance from Berlin to Rome as c.

Therefore we say

a = 697miles, b = 545miles, c = 734miles

To find the respectively angle formed from, Berlin to Paris to Rome, we will apply the cosine rule, which states that:

c² = a² + b² - 2abCosC

734² = 697² + 545² - 2(697*545)cosC

2(697*545)cosC = 697² + 545² - 734²

759739 cosC = 244078

cosC = 244078/759739

cosC = 0.3212

C = cos-¹0.3212

C = 71.26°

To get angle B, we will apply the sine rule, which states that:

a/sineA = b/sineB = c/sineC

Therefore we have

545/sineB = 734/sine71.26

= 545/sineB = 734/0.947

= 545/sineB = 734/0.947

= 545/sineB = 775.08

SineB = 545/775.08

B = sine-¹0.7032

B = 44.68°

Angle A = 180 - (B+C) angle in a triangle

A = 180 - (44.68 + 71.26)

A = 180 - 115.94

A = 64.06°

A = 64.06°

B = 44.68°

C = 71.26°

Angle Rome to Paris = 44.68°

Angle Paris to Berlin = 71.26°

Angle Berlin to Rome = 64.06°