Problem 44 A triangular parcel of land has ... [FREE SOLUTION]

Chapter 5: Problem 44

A triangular parcel of land has 115 meters of frontage, and the other boundaries have lengths of 76 meters and 92 meters. What angles does the frontage make with the two other boundaries?

Short Answer

Expert verified

To find the angles the frontage makes with the other boundaries, apply the law of cosines and solve for each angle

Step by step solution

Identifying Given Information

Identify the given information, which are the sides of the triangle. Let's call the frontage \(a\) and the other two boundaries \(b\) and \(c\). So, \(a = 115m\), \(b = 76m\) and \(c = 92m\). Our task is to find the angles between \(a\), \(b\) and \(c\), let's denote them as \(\alpha\), \(\beta\) and \(\gamma\). \(\alpha\) is the angle between \(b\) and \(c\), \(\beta\) is the angle between \(a\) and \(c\), and \(\gamma\) is the angle between \(a\) and \(b\).

Apply Law of Cosines

The law of cosines states that for any triangle with sides \(a\), \(b\), \(c\) and respective opposite angles \(\alpha\), \(\beta\), \(\gamma\), the following equations hold true: \(a^2 = b^2 + c^2 - 2bc \cos(\alpha)\), \(b^2 = a^2 + c^2 - 2ac \cos(\beta)\), \(c^2 = a^2 + b^2 - 2ab \cos(\gamma)\). The angles can be obtained by rearranging these equations. For \(\alpha\), \(\cos(\alpha) = (b^2 + c^2 - a^2) / 2bc\). For \(\beta\), \(\cos(\beta) = (a^2 + c^2 - b^2) / 2ac\). For \(\gamma\), \(\cos(\gamma) = (a^2 + b^2 - c^2) / 2ab\).

Calculate the Measures of the Angles

Use the above formulas from Step 2 to calculate the measures of angles \(\alpha\), \(\beta\) and \(\gamma\). Make sure to convert the angles from radians to degrees by multiplying by \((180/ \pi)\). Do remember that, the output of the cos function will be in radians and needs to be converted to degrees. Also, the result must be passed through the inverse cosine function (if the radian value is x then the degree value will be \(\cos^{-1}(x)\))

Validate the Results

Validate the results by checking that they satisfy the definition of a triangle - the sum of all angles in a triangle is 180 degrees. Sum up \(\alpha\), \(\beta\) and \(\gamma\) and ensure they add up to 180 degrees

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91影视

A triangular parcel of land has 115 meters of frontage, and the other boundaries have lengths of 76 meters and 92 meters. What angles does the frontage make with the two other boundaries?

Short Answer

Step by step solution

Identifying Given Information

Apply Law of Cosines

Calculate the Measures of the Angles

Validate the Results

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