Problem 16 - Algebra 2

Question

Find the area of \(\triangle A B C\) to the nearest tenth. $$ B=32^{\circ}, a=11 \mathrm{mi}, c=5 \mathrm{mi} $$

Step-by-Step Solution

Answer

The area of \(\triangle ABC\) is approximately 14.6 square miles.

Step 1: Identify Known Values

In the given triangle, we know the length of side \(a = 11\) miles, side \(c = 5\) miles, and angle \(B = 32^{\circ}\). We will use these values to find the area of \(\triangle ABC\).

Step 2: Use the Formula for Area of a Triangle

To find the area of the triangle, we use the formula for area with two sides and an included angle: \( \text{Area} = \frac{1}{2} a c \sin B \). Substitute the known values into the formula.

Step 3: Calculate \(\sin 32^{\circ}\)

Calculate \(\sin 32^{\circ}\) using a calculator: \(\sin 32^{\circ} \approx 0.5299\).

Step 4: Substitute and Compute the Area

Now substitute \(a = 11\), \(c = 5\), and \(\sin 32^{\circ}\) into the area formula: \( \text{Area} = \frac{1}{2} \times 11 \times 5 \times 0.5299 \).

Step 5: Calculate the Final Area

Perform the calculation: \( \text{Area} = \frac{1}{2} \times 11 \times 5 \times 0.5299 = 14.5725 \). Round this result to the nearest tenth.

Step 6: Round the Area to the Nearest Tenth

The calculated area is \(14.5725\). When rounded to the nearest tenth, the area is \(14.6\) square miles.

Key Concepts

Law of Sines

The Law of Sines is one of the fundamental tools in triangle geometry, especially when dealing with non-right triangles. It states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is constant for all three sides and angles in the triangle. This can be mathematically represented as follows:

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

In our problem, although we mainly used the sine function to find the area, the Law of Sines is particularly useful when you need to find unknown sides or angles. For instance, given one angle and two sides, you can find another angle of the triangle. This systematic approach often parallels with the calculation of areas, where the sine of specific angles play pivotal roles. Understanding the Law of Sines enhances one's ability to work with and visualize triangle content effectively.

Trigonometric Ratios

Trigonometric ratios are core to solving many types of problems involving triangles, including finding their areas. In any triangle, these ratios relate the angles and lengths of sides, providing powerful relationships and formulas. The three primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan), which correspond to different sides of right-angled triangles.

• Sine, as used in this exercise, is defined for a right triangle as the ratio of the length of the opposite side to the hypotenuse. For any angle \( \theta \), \( \sin\theta = \frac{\text{Opposite side}}{\text{Hypotenuse}} \).

In this particular problem, since we were dealing with an oblique triangle (a triangle without a right angle), using the sine function was crucial. Thus, to find the area of \(\triangle ABC\), knowing \(\sin 32^{\circ}\) was central. Calculators or trigonometric tables can help in finding these values, allowing precise calculations involving triangle properties.

Triangle Geometry

Triangle geometry is a wide-ranging topic encompassing various methods and formulas to understand and solve problems related to triangles. A triangle is defined by its three sides and three angles. The sum of all internal angles in any triangle is always \(180^{\circ}\).When calculating areas, several formulas exist depending on the information available. For instance, the basic formula for the area is \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \), but if given two sides and an included angle, the formula becomes \( \text{Area} = \frac{1}{2} a c \sin B \), as demonstrated in this problem.The understanding of these geometric concepts facilitates solving complex problems, providing methods for proving properties or constructing triangles based on specific conditions. Grasping these fundamental elements helps in both academic and real-world applications, such as architecture or engineering.