Problem 24 - Algebra 2 and Trigonometry

Question

A small park is in the shape of an isosceles trapezoid. The length of the longer of the parallel sides is 3.2 kilometers and the length of an adjacent side is 2.4 kilometers. A path from one corner of the park to an opposite corner is 3.6 kilometers long. a. Find, to the nearest tenth, the measure of each angle between adjacent sides of the park. b. Find, to the nearest tenth, the measure of each angle between the path and a side of the park. c. Find, to the nearest tenth, the length of the shorter of the parallel sides.

Step-by-Step Solution

Answer

Angles: 78.5° and 101.5°, Shorter base: 2.1 km.

Step 1: Understand the Isosceles Trapezoid

An isosceles trapezoid has two parallel sides, which are known as the bases, and two non-parallel sides that are equal in length. Given data: longer base = 3.2 km, adjacent side = 2.4 km (both sides are equal), and the path (diagonal) = 3.6 km.

Step 2: Draw a Triangle Within the Trapezoid

Visualize or sketch a triangle by connecting the ends of the longer base to one of the non-parallel sides, forming a triangle with one base, one non-parallel side, and the diagonal as sides.

Step 3: Use the Law of Cosines for the Triangle

Apply the law of cosines in the triangle with sides of 3.2 km (base), 2.4 km (side), and 3.6 km (diagonal): \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \] Plug the values into the formula: \[ 3.6^2 = 3.2^2 + 2.4^2 - 2 \cdot 3.2 \cdot 2.4 \cdot \cos(C) \] Calculate to get \( C \), the angle between the base (3.2 km) and the side (2.4 km).

Step 4: Calculate the Angle Using the Law of Cosines

Compute the cosine:\[ 12.96 = 10.24 + 5.76 - 15.36 \cdot \cos(C) \]\[ 12.96 = 16 - 15.36 \cdot \cos(C) \]\[ \cos(C) = \frac{16 - 12.96}{15.36} \]Solve for \( C \) using a calculator:\[ C \approx \cos^{-1}(0.198) \approx 78.5^{\circ} \]

Step 5: Angle Complements in an Isosceles Triangle

Each angle between the longer base and an adjacent side in an isosceles trapezoid is the same: 78.5°. Since these angles sum up to 180° with the angle opposite the longer base (the path angle), the angle between the path and the side at each non-parallel vertex is 101.5°.

Step 6: Use the Sine Rule to Find the Shorter Base

In the triangle formed by the path, apply the law of sines to find the angle opposite to the shorter base (3.6 km diagonal): \[ \frac{3.6}{\sin(78.5^{\circ})} = \frac{x}{\sin(101.5^{\circ})} \] Solve for \(x\): \[ x = \frac{3.6 \cdot \sin(101.5^{\circ})}{\sin(78.5^{\circ})} \]

Step 7: Calculate and Answer the Questions

Calculate the length of the shorter base using a calculator: \[ x \approx 2.1 \text{ km} \]Summary:- Angles between adjacent sides and path: 78.5° and 101.5°.- Shorter base length: 2.1 km.

Key Concepts

Law of Cosines

When you encounter a problem involving triangles, particularly one that needs calculating the angles or sides, the Law of Cosines is incredibly helpful. This law is derived from extending the Pythagorean theorem to non-right angled triangles. It is given by the formula:

• \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]

Here, \( c \) is the side opposite the angle you are trying to find, \( a \) and \( b \) are the other two sides, and \( C \) is the angle opposite \( c \). In the case of the isosceles trapezoid problem, this formula allowed us to solve for the angle \( C \) by rearranging it to find \( \cos(C) \), and then calculating \( C \) using an inverse cosine function.

The applicability of the Law of Cosines extends to instances where two sides and the included angle are known (SAS), or when all three sides are known (SSS) to find an angle. Calculating the angle in this isosceles trapezoid made finding other necessary elements of the trapezoid much easier.

Isosceles Trapezoid Properties

Isosceles trapezoids are a fascinating geometric shape with distinctive properties. They are defined by having a pair of opposite sides that are parallel, commonly referred to as the bases, and the other two sides being equal in length, known as the legs. This symmetrical configuration leads to unique angle relationships and can simplify calculations.

• The angles adjacent to each base in an isosceles trapezoid are equal due to the symmetry of the legs.
• The diagonals of an isosceles trapezoid also possess interesting traits: they are of equal length and bisect one another.

In our problem, understanding that the sides are equal in length was essential in setting up the triangle and using trigonometric principles to find the unknown angles. Moreover, knowing the sum of the angles between the non-parallel sides and the base (those adjacent to the longer base) helped deduce the angle at the opposite vertex, revealing more about the trapezoid's geometry.

Sine Rule

The Sine Rule is another powerful tool in trigonometry that helps solve triangle problems by relating the sides of the triangle to their opposite angles. It's given by the relationship:

• \[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \]

This formula is particularly handy when dealing with non-right triangles where a side and its opposite angle are known, as well as another angle. It can be used to determine an unknown side or angle in the triangle.

In the context of the isosceles trapezoid problem, the Sine Rule enabled us to find the length of the shorter base. By knowing the diagonal (path) and using the calculated angles adjacent to it, we could set up an equation that solved for the missing base length. This technique underscores the versatility of the Sine Rule in bridging gaps when information about certain sides or angles in a triangle is limited.