Triangle Perimeter, Area & Angles: Step-by-Step Guide

Hey guys! Today, we're diving into a fun geometry problem: figuring out the perimeter, area, and internal angles of a triangle. We've got the vertices A(-4, 2), B(6, 5), and C(3, -5). Plus, we'll sketch it out on the Cartesian plane and show those handy auxiliary lines for angle calculations. Let's get started!

1. Plotting the Triangle on the Cartesian Plane

First things first, let’s visualize our triangle. Grab your graph paper (or your favorite digital graphing tool) and plot the points A(-4, 2), B(6, 5), and C(3, -5). Connect the dots, and boom! You’ve got your triangle ABC. Seeing it visually helps us understand the problem better. Plotting these points accurately is crucial because the visual representation will aid in understanding the relationships between the sides and angles of the triangle. You can easily identify the rough shape of the triangle, which will help you anticipate the nature of the angles and the relative lengths of the sides. This visual aid is invaluable for double-checking your calculations later on. For instance, if you calculate an angle to be obtuse but the triangle appears to be acute on the graph, you know there might be an error in your calculations. Similarly, the visual representation will help you in estimating the lengths of the sides, making it easier to catch mistakes in the distance formula calculations.

2. Calculating the Side Lengths (Perimeter Prep)

The perimeter is simply the sum of the lengths of all three sides. To find these lengths, we'll use the distance formula:

• Distance = √((x₂ - x₁)² + (y₂ - y₁)²)

Let's calculate each side:

• AB: √((6 - (-4))² + (5 - 2)²) = √(10² + 3²) = √109 ≈ 10.44 units
• BC: √((3 - 6)² + (-5 - 5)²) = √((-3)² + (-10)²) = √109 ≈ 10.44 units
• CA: √((-4 - 3)² + (2 - (-5))²) = √((-7)² + 7²) = √98 = 7√2 ≈ 9.90 units

Notice that sides AB and BC have the same length, meaning our triangle is isosceles. This is a great observation, as it gives us a head start in understanding the triangle's properties. Calculating the lengths of the sides is an essential step, not only for finding the perimeter but also for later calculations such as the area and angles. By finding these lengths accurately, you lay the groundwork for the rest of the problem. The distance formula is a direct application of the Pythagorean theorem, which gives the straight-line distance between two points in a coordinate plane. This method is accurate and universally applicable, regardless of the orientation or position of the triangle in the plane. The numerical values obtained for the side lengths allow you to make comparisons and draw conclusions about the shape and type of the triangle, as we did when we noticed that two sides are equal.

3. Finding the Perimeter

Now that we have the side lengths, the perimeter is a breeze:

• Perimeter = AB + BC + CA ≈ 10.44 + 10.44 + 9.90 ≈ 30.78 units

So, the perimeter of triangle ABC is approximately 30.78 units. The perimeter is a fundamental property of any polygon, representing the total distance around its exterior. It’s a straightforward calculation, but it is crucial in many applications, from fencing a yard to calculating the amount of material needed for a border. In the context of our triangle, knowing the perimeter gives us an overall sense of the triangle's size. Additionally, if we were dealing with a real-world problem, such as finding the distance a runner would cover running around the triangular path, the perimeter would be exactly the information we need. The perimeter can also be used as a preliminary check in other calculations. For instance, if you were to use Heron’s formula to find the area of the triangle, you would need the semi-perimeter, which is simply half of the perimeter. Therefore, accurately calculating the perimeter is not just an end in itself but also a useful step in more complex calculations.

4. Calculating the Area (Using Heron's Formula)

For the area, we’ll use Heron's formula since we know all three side lengths. First, we need the semi-perimeter (s):

• s = Perimeter / 2 ≈ 30.78 / 2 ≈ 15.39 units

Heron's formula states:

• Area = √(s(s - AB)(s - BC)(s - CA))
• Area ≈ √(15.39(15.39 - 10.44)(15.39 - 10.44)(15.39 - 9.90))
• Area ≈ √(15.39 * 4.95 * 4.95 * 5.49)
• Area ≈ √2074.47
• Area ≈ 45.55 square units

Thus, the area of triangle ABC is approximately 45.55 square units. Heron’s formula is particularly useful because it allows us to calculate the area of a triangle using only the lengths of its sides, without needing to know any angles or the height. This is a significant advantage in situations where finding the height directly might be complicated. The semi-perimeter, a key component of Heron's formula, represents half the distance around the triangle and helps simplify the calculation. By taking the square root of the product of the semi-perimeter and the differences between the semi-perimeter and each side, we arrive at the area. This formula is a testament to the elegance and utility of mathematical relationships, allowing us to derive a fundamental property of the triangle from just a few basic measurements. Moreover, the area is an essential parameter in many applications, from determining the amount of paint needed to cover a triangular surface to calculating the load-bearing capacity of a triangular structural element.

5. Determining the Internal Angles (Using the Law of Cosines)

To find the internal angles, we can use the Law of Cosines. Let's find angle A first:

• BC² = AB² + CA² - 2(AB)(CA)cos(A)
• 109 = 109 + 98 - 2(10.44)(9.90)cos(A)
• 0 = 98 - 206.71cos(A)
• cos(A) = 98 / 206.71
• cos(A) ≈ 0.474
• A ≈ arccos(0.474) ≈ 61.72 degrees

Now, let's find angle B:

• CA² = AB² + BC² - 2(AB)(BC)cos(B)
• 98 = 109 + 109 - 2(10.44)(10.44)cos(B)
• 98 = 218 - 217.99cos(B)
• 217.99cos(B) = 120
• cos(B) ≈ 120 / 217.99
• cos(B) ≈ 0.550
• B ≈ arccos(0.550) ≈ 56.64 degrees

Finally, we can find angle C by subtracting angles A and B from 180 degrees:

• C = 180 - A - B ≈ 180 - 61.72 - 56.64 ≈ 61.64 degrees

So, our angles are approximately: ∠A ≈ 61.72 degrees, ∠B ≈ 56.64 degrees, and ∠C ≈ 61.64 degrees. Isn't this cool? The Law of Cosines is a powerful tool for finding angles in any triangle when you know the lengths of all three sides. It’s a generalization of the Pythagorean theorem, which applies only to right triangles. The Law of Cosines can be used for any triangle, making it a versatile tool in trigonometry. By rearranging the formula, we can isolate the cosine of the angle we want to find and then use the inverse cosine function (arccos) to get the angle in degrees. Accurately calculating angles is crucial in various fields, from engineering and navigation to computer graphics and physics. In the context of this problem, knowing the angles allows us to fully characterize the triangle and verify the consistency of our calculations. For example, the sum of the internal angles of a triangle must always be 180 degrees, so we can use this as a check for our results.

6. Drawing Auxiliary Lines (If Needed)

For this particular problem, drawing auxiliary lines isn't strictly necessary since we’ve successfully calculated everything using formulas. However, if we were using a more geometric approach to find the angles, auxiliary lines (like altitudes or angle bisectors) could help create similar triangles or right triangles, making the calculations easier. In some cases, auxiliary lines are invaluable for solving complex geometric problems. They create new relationships and structures within the figure, which can lead to a solution that might not be immediately obvious. For example, drawing an altitude from one vertex to the opposite side can create two right triangles, allowing you to use trigonometric ratios or the Pythagorean theorem. Similarly, angle bisectors can create congruent triangles, enabling you to establish relationships between angles and sides. The key to using auxiliary lines effectively is to choose them strategically, based on the specific properties and requirements of the problem. Practice and familiarity with geometric theorems and constructions will help you develop the intuition needed to make these choices.

7. Recap and Verification

Let's recap our findings:

• Perimeter: ≈ 30.78 units
• Area: ≈ 45.55 square units
• Angles: ∠A ≈ 61.72 degrees, ∠B ≈ 56.64 degrees, ∠C ≈ 61.64 degrees

A quick check: 61.72 + 56.64 + 61.64 ≈ 180 degrees (Yay!)

Conclusion

And there you have it! We've successfully determined the perimeter, area, and internal angles of triangle ABC, and even sketched it on the Cartesian plane. Geometry can be a blast when you break it down step by step. Keep practicing, and you’ll become a triangle-solving pro in no time! Hope this helped, and happy calculating, everyone!