Calculating The Area Of A Right Triangle A Step-by-Step Guide

In the realm of geometry, right triangles hold a special place. Their unique properties make them fundamental building blocks for various calculations and applications. This comprehensive guide will walk you through the process of determining the area of a right triangle, given specific information about its angles and side lengths. We'll tackle a particular problem, breaking it down into manageable steps, and provide explanations along the way to ensure a clear understanding of the concepts involved. The problem we'll be addressing is: A right triangle has one angle that measures $23^{\circ}$. The adjacent leg measures 27.6 cm, and the hypotenuse measures 30 cm. What is the approximate area of the triangle? Round to the nearest tenth.

Understanding the Fundamentals of Right Triangles

Before diving into the calculations, let's establish a solid foundation by revisiting the key characteristics of right triangles. A right triangle is defined as a triangle that contains one interior angle measuring exactly $90^{\circ}$. This angle is often referred to as the right angle. The side opposite the right angle is the longest side of the triangle and is called the hypotenuse. The other two sides, which form the right angle, are called legs or cathetus. One of the legs is considered the adjacent leg, and the other is the opposite leg, relative to a specific acute angle (an angle less than $90^{\circ}$) within the triangle.

The area of any triangle is calculated using the formula: Area = $\frac{1}{2}$ * base * height. In a right triangle, the two legs naturally serve as the base and height, simplifying the area calculation. Therefore, the area of a right triangle can be expressed as: Area = $\frac{1}{2}$ * leg1 * leg2. It is crucial to remember this formula as it forms the basis for solving our problem. Moreover, understanding trigonometric ratios becomes essential when dealing with angles and side lengths in right triangles. The primary trigonometric ratios are sine (sin), cosine (cos), and tangent (tan), which relate the angles of a right triangle to the ratios of its sides. These ratios are defined as follows:

• Sine (sin) of an angle = (Opposite side) / (Hypotenuse)
• Cosine (cos) of an angle = (Adjacent side) / (Hypotenuse)
• Tangent (tan) of an angle = (Opposite side) / (Adjacent side)

These ratios are invaluable tools for determining unknown side lengths or angles within a right triangle, given sufficient information. In our specific problem, we are provided with the measure of one acute angle ($23^{\circ}$), the length of the adjacent leg (27.6 cm), and the length of the hypotenuse (30 cm). Our goal is to calculate the area of the triangle, which requires us to know the lengths of both legs. Since we already have the length of the adjacent leg, we need to find the length of the opposite leg. This is where trigonometric ratios come into play. We can use the sine function, which relates the opposite side to the hypotenuse, or the tangent function, which relates the opposite side to the adjacent side. The choice depends on the information we have available and which ratio will allow us to directly solve for the unknown side length. Let's explore how we can apply these concepts to solve the problem at hand.

Step-by-Step Solution to Find the Triangle's Area

Now, let's tackle the problem of finding the area of the given right triangle. We'll break down the solution into a series of clear steps, ensuring that each step is well-explained and easy to follow. By systematically addressing the problem, we can avoid confusion and arrive at the correct answer with confidence.

Step 1: Identify the Given Information

Before we begin any calculations, it's essential to clearly identify the information provided in the problem statement. This step helps us to organize our thoughts and determine the best approach for solving the problem. In this case, we are given the following:

• One angle measures $23^{\circ}$.
• The adjacent leg measures 27.6 cm.
• The hypotenuse measures 30 cm.

We are also reminded of the formula for the area of a triangle: Area = $\frac{1}{2}$ * base * height. In a right triangle, the legs serve as the base and height. Therefore, to find the area, we need to determine the lengths of both legs.

Step 2: Determine the Missing Leg Length

We already know the length of the adjacent leg (27.6 cm). To calculate the area, we need to find the length of the opposite leg. We can use trigonometric ratios to relate the given angle and side lengths to the unknown side length. Specifically, we can use the sine function or the tangent function. Let's explore both approaches.

• Using the Sine Function:

The sine of an angle is defined as the ratio of the opposite side to the hypotenuse. In our case, we have: sin($23^{\circ}$) = (Opposite leg) / (Hypotenuse). We know the hypotenuse is 30 cm, so we can write: sin($23^{\circ}$) = (Opposite leg) / 30 cm. To solve for the opposite leg, we multiply both sides of the equation by 30 cm: Opposite leg = 30 cm * sin($23^{\circ}$). Using a calculator, we find that sin($23^{\circ}$) ≈ 0.3907. Therefore, the opposite leg is approximately 30 cm * 0.3907 ≈ 11.72 cm.

• Using the Tangent Function:

The tangent of an angle is defined as the ratio of the opposite side to the adjacent side. In our case, we have: tan($23^{\circ}$) = (Opposite leg) / (Adjacent leg). We know the adjacent leg is 27.6 cm, so we can write: tan($23^{\circ}$) = (Opposite leg) / 27.6 cm. To solve for the opposite leg, we multiply both sides of the equation by 27.6 cm: Opposite leg = 27.6 cm * tan($23^{\circ}$). Using a calculator, we find that tan($23^{\circ}$) ≈ 0.4245. Therefore, the opposite leg is approximately 27.6 cm * 0.4245 ≈ 11.72 cm.

As we can see, both methods yield the same result for the length of the opposite leg, which is approximately 11.72 cm. This confirms the consistency of our approach and reinforces our confidence in the solution.

Step 3: Calculate the Area of the Triangle

Now that we have the lengths of both legs (27.6 cm and 11.72 cm), we can easily calculate the area of the triangle using the formula: Area = $\frac{1}{2}$ * base * height. In our case, the legs serve as the base and height, so we have: Area = $\frac{1}{2}$ * 27.6 cm * 11.72 cm. Multiplying these values, we get: Area ≈ $\frac{1}{2}$ * 323.472 cm². Dividing by 2, we find the area to be approximately 161.736 cm².

Step 4: Round to the Nearest Tenth

The problem asks us to round the area to the nearest tenth. Looking at our calculated area of 161.736 cm², the digit in the tenths place is 7, and the digit in the hundredths place is 3. Since 3 is less than 5, we round down, keeping the digit in the tenths place as 7. Therefore, the approximate area of the triangle, rounded to the nearest tenth, is 161.7 cm².

Conclusion: Mastering Right Triangle Area Calculations

In this comprehensive guide, we've successfully navigated the process of calculating the area of a right triangle, given specific information about its angles and side lengths. We started by establishing a solid understanding of the fundamental properties of right triangles, including the definitions of the hypotenuse, legs, and trigonometric ratios. We then tackled a specific problem, breaking it down into manageable steps and providing detailed explanations along the way. By systematically identifying the given information, determining the missing leg length using trigonometric functions, calculating the area using the appropriate formula, and rounding to the nearest tenth, we arrived at the accurate solution of 161.7 cm².

This exercise demonstrates the importance of a clear understanding of geometric principles and the application of trigonometric ratios in solving real-world problems. By mastering these concepts, you can confidently approach a wide range of geometric challenges and develop a deeper appreciation for the elegance and power of mathematics. Remember, practice is key to solidifying your understanding and building your problem-solving skills. So, continue to explore various geometric problems and challenge yourself to apply the techniques you've learned. With consistent effort and a solid foundation, you'll be well-equipped to excel in geometry and beyond.