Problem 42 - Algebra 2

Question

Measurement A vacant lot is in the shape of an isosceles triangle. It is between two streets that intersect at an \(85.9^{\circ}\) angle. Each of the sides of the lot that face these streets is 150 \(\mathrm{ft}\) long. Find the length of the third side, to the nearest foot.

Step-by-Step Solution

Answer

The length of the third side, rounded to the nearest foot, is obtained by following these steps.

Step 1: Understanding the Law of Cosines

The Law of Cosines can be stated as \(c^2 = a^2 + b^2 - 2ab \cdot cos(C)\) , where \(a\), \(b\), and \(c\) are the sides of a triangle, and \(C\) is the angle opposite to side \(c\). In our case, the sides are 150ft, 150ft, and the third side (unknown), while the angle is \(85.9^{\circ}\).

Step 2: Apply the Law of Cosines

Replace the known values in the Law of Cosines formula. We have:\(c^2 = (150 ft)^2 + (150 ft)^2 - 2 * 150 ft * 150 ft * cos(85.9^{\circ})\)

Step 3: Calculate the Square of the Third Side

After replacing the values and doing the calculations, you obtain a numerical value for the square of the length of the third side. It includes the cosine of \(85.9^{\circ}\), which can be easily calculated using a scientific calculator.

Step 4: Find the Third Side

The value obtained from the above step gives the square of the length of the third side. To find the actual length of the side, take the square root of this result. Also, as required by the problem, round the result to the nearest foot.

Key Concepts

Understanding Isosceles Triangles

An isosceles triangle is a unique type of triangle where two of its sides are equal in length. This feature results in two angles within the triangle also being equal. These equal angles are opposite the equal sides. In the original problem, the vacant lot is in the shape of such a triangle.

• The sides facing the streets are both 150 feet long, forming the equal sides of the triangle.
• The angle between these two sides is 85.9 degrees.
• The length of the third side is unknown and needs to be calculated.

Understanding the properties of an isosceles triangle helps us to use specific methods and formulas, such as the Law of Cosines, to efficiently calculate unknown elements of the triangle.

The Basics of Angle Measurement

The measurement of angles is a fundamental part of working with triangles and other geometric shapes. Angles are usually measured in degrees, with a full circle measuring 360 degrees. In the given scenario, the problem provides an angle measurement of 85.9 degrees.

• This angle is crucial since it is located between the two known sides of the isosceles triangle.
• When determining lengths of sides using trigonometric formulas, angle measurements need to be accurate for the calculations to be precise.
• Tools like protractors or scientific calculators can help in measuring or computing the cosine of angles when working through problems like this one.

Mastering angle measurement is key to solving many geometrical problems, especially when they involve calculating unknown sides or angles using trigonometric laws.

Calculating Triangle Sides Using the Law of Cosines

In any triangle, when you know two sides and the included angle (like in our isosceles triangle), the Law of Cosines is a powerful formula to find the unknown third side. The formula is stated as \( c^2 = a^2 + b^2 - 2ab \cos(C) \), where \( a \) and \( b \) are the known sides, \( C \) is the included angle, and \( c \) is the unknown side.

• For this particular problem, substitute 150 feet for both \( a \) and \( b \), and 85.9 degrees for \( C \).
• The calculation involves squaring the known sides and using the cosine of the angle.
• A scientific calculator is typically needed to determine the cosine of 85.9 degrees accurately.

Once the computations are complete, the result will provide the square of the unknown side. Taking the square root of this result gives the length of the third side. Always remember to round to the nearest foot for precision in practical applications.

Employing Trigonometry in Geometrical Problems

Trigonometry is a branch of mathematics dealing with the relationships between the sides and angles of triangles. It is incredibly useful in solving a wide variety of problems related to angles and lengths in geometrical figures. In this exercise, the Law of Cosines originates from trigonometry and is instrumental in finding the missing side of the isosceles triangle.

• The cosine function, one of the primary trigonometric functions, is crucial in the Law of Cosines.
• Being comfortable with using trigonometric functions like sine, cosine, and tangent can significantly aid in solving complex problems.
• Real-world scenarios often require accurate angle and side measurements, which is where trigonometric principles shine.

Understanding trigonometry enhances problem-solving skills and opens up many opportunities for application in fields such as engineering, physics, and architecture.