Calculating Triangle Area A Step-by-Step Guide

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Hey guys! Today, we're diving into a fun geometry problem that involves calculating the area of a right triangle. We'll break it down step-by-step, so even if math isn't your favorite subject, you'll be able to follow along. Let's get started!

Understanding the Problem

So, here's the scenario: We have a right triangle, which means one of its angles is exactly 90 degrees. We know one of the other angles measures 23 degrees. The leg adjacent to this 23-degree angle (the side next to it that's not the hypotenuse) is 27.6 cm long. And the hypotenuse, which is always the longest side in a right triangle and opposite the right angle, measures 30 cm. Our mission, should we choose to accept it (and we do!), is to find the approximate area of this triangle, rounded to the nearest tenth.

The area of a triangle is calculated using the formula: $Area = \frac{1}{2} * base * height$. In a right triangle, the two legs (the sides that form the right angle) can be considered the base and height. We already know the length of one leg (the adjacent leg), but we need to find the length of the other leg (the opposite leg) to calculate the area. Don't worry, we'll use some trigonometry magic to figure that out!

Using Trigonometry to Find the Missing Side

Remember SOH CAH TOA? It's a handy acronym that helps us remember the trigonometric ratios. It stands for:

  • SOH: Sine = Opposite / Hypotenuse
  • CAH: Cosine = Adjacent / Hypotenuse
  • TOA: Tangent = Opposite / Adjacent

In our case, we know the adjacent side (27.6 cm) and the hypotenuse (30 cm), and we want to find the opposite side. So, which trigonometric ratio should we use? If you guessed tangent (TOA), you're on the right track! However, since we don't know the angle opposite the side we're trying to find, and we do know the hypotenuse, we need to use either sine or cosine. Since we know the adjacent side and the hypotenuse, we'll use the cosine (CAH). This is a crucial step, guys, so make sure you're following along. We're essentially using the information we have (the adjacent side and the hypotenuse) along with the known angle to uncover the missing piece of the puzzle – the opposite side, which will serve as our height when calculating the area.

Let's set up the equation: $cos(23^{\circ}) = \frac{Adjacent}{Hypotenuse} = \frac{27.6}{30}$. This equation relates the cosine of the 23-degree angle to the ratio of the adjacent side (which we know) to the hypotenuse (which we also know). However, this equation doesn't directly help us find the opposite side. We need a different approach. The key realization here is that we need to use a trigonometric ratio that involves the opposite side directly. That's where the sine function (SOH) comes in. While we used the cosine to initially understand the relationship between the given sides and the angle, the sine function is our ticket to finding the missing side length required for the area calculation.

To find the length of the opposite side, we need to use the sine function. We first need to find the other angle in the triangle (other than the right angle and the 23-degree angle). Since the angles in a triangle add up to 180 degrees, and one angle is 90 degrees, the other angle is $180 - 90 - 23 = 67$ degrees. Now we can use the sine of this angle:

sin(67∘)=OppositeHypotenusesin(67^{\circ}) = \frac{Opposite}{Hypotenuse}

Now, let's plug in the values we know:

sin(67∘)=Opposite30sin(67^{\circ}) = \frac{Opposite}{30}

To solve for the opposite side, we multiply both sides of the equation by 30:

Opposite=30∗sin(67∘)Opposite = 30 * sin(67^{\circ})

Using a calculator, we find that $sin(67^{\circ}) \approx 0.9205$, so:

Opposite≈30∗0.9205≈27.615cmOpposite \approx 30 * 0.9205 \approx 27.615 cm

This calculation is super important because it gives us the height of the triangle. Remember, the height is crucial for calculating the area. We've used the sine function, a bit of angle manipulation, and some basic algebra to uncover this missing dimension. Now that we have both the base (the adjacent side) and the height (the opposite side), we're just a quick formula application away from the final answer.

Calculating the Area

Now that we've found the length of the opposite side (approximately 27.615 cm), we can finally calculate the area of the triangle. Remember the formula: $Area = \frac{1}{2} * base * height$.

We know the base is 27.6 cm and the height is approximately 27.615 cm. Plugging these values into the formula, we get:

Area=12∗27.6∗27.615Area = \frac{1}{2} * 27.6 * 27.615

Let's do the math:

Area≈12∗761.174≈380.587cm2Area \approx \frac{1}{2} * 761.174 \approx 380.587 cm^2

The final step is to round the area to the nearest tenth, as the problem asked. So, 380.587 rounded to the nearest tenth is 380.6. This rounding is essential for providing the answer in the correct format and level of precision. It demonstrates attention to detail and ensures that we're meeting the specific requirements of the problem.

The Answer

Therefore, the approximate area of the triangle is 380.6 square centimeters. Great job, guys! We successfully navigated through the trigonometric relationships, side calculations, and the area formula to arrive at the solution. This problem beautifully illustrates how different geometric concepts intertwine and how we can use them together to solve complex problems.

Key Takeaways

  • Right triangles are special because we can use trigonometric ratios (SOH CAH TOA) to find missing sides and angles.
  • The area of a triangle is calculated as half the base times the height.
  • Trigonometric functions like sine and cosine are essential tools for solving geometry problems involving triangles.
  • Careful rounding is important to provide answers in the correct format.

So, there you have it! We've conquered another math problem. Keep practicing, and you'll become a geometry whiz in no time!