Triangle ABC: Calculating Perimeter And Area

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Hey guys! Let's dive into a classic geometry problem. We're given a triangle, aptly named ABC, and we have the coordinates of its vertices: A(2, 8), B(16, 2), and C(6, 2). Our mission, should we choose to accept it (and we totally will!), is to figure out the perimeter and the area of this triangle. Sounds like fun, right? Don't worry, it's not as scary as it might seem. We'll break it down step by step, using some fundamental concepts of coordinate geometry. By the end, you'll be a triangle-solving pro! We're going to use the distance formula to find the lengths of the sides and then use a couple of different methods to calculate the area. Get ready to flex those math muscles!

Understanding the Basics: Coordinates and the Distance Formula

Okay, before we jump into the nitty-gritty, let's make sure we're all on the same page. Remember those coordinate planes from middle school? Each point on the plane is defined by an (x, y) coordinate pair. So, A(2, 8) means that the point A is located 2 units to the right on the x-axis and 8 units up on the y-axis. B(16, 2) and C(6, 2) follow the same logic. Now, to find the lengths of the sides of our triangle, we'll need the distance formula. This formula is our secret weapon for calculating the distance between two points in a coordinate plane.

The distance formula is derived from the Pythagorean theorem, so it's all interconnected! For two points, (x1, y1) and (x2, y2), the distance (d) between them is calculated as follows:

d = √((x2 - x1)² + (y2 - y1)²)

Basically, we find the difference in the x-coordinates, square it, find the difference in the y-coordinates, square it, add those two results together, and then take the square root. Bam! We have the distance. Let's get down to the business of calculation! It seems a bit complicated at first glance, but once you start to use it, it becomes second nature, I swear!

Finding the Lengths of the Sides

Alright, let's apply the distance formula to our triangle ABC. We need to find the lengths of AB, BC, and AC.

  1. Length of AB:

    • A(2, 8), B(16, 2)
    • AB = √((16 - 2)² + (2 - 8)²)
    • AB = √((14)² + (-6)²)
    • AB = √(196 + 36)
    • AB = √232
    • AB ≈ 15.23 units

  2. Length of BC:

    • B(16, 2), C(6, 2)
    • BC = √((6 - 16)² + (2 - 2)²)
    • BC = √((-10)² + (0)²)
    • BC = √100
    • BC = 10 units

  3. Length of AC:

    • A(2, 8), C(6, 2)
    • AC = √((6 - 2)² + (2 - 8)²)
    • AC = √((4)² + (-6)²)
    • AC = √(16 + 36)
    • AC = √52
    • AC ≈ 7.21 units

See? Not so bad, right? We've successfully calculated the lengths of all three sides of our triangle using the distance formula. Now, we're ready to find the perimeter.

Calculating the Perimeter of Triangle ABC

Alright, now that we've found the lengths of all the sides, calculating the perimeter is a piece of cake. The perimeter of any polygon is simply the sum of the lengths of all its sides. For our triangle ABC, this means adding the lengths of AB, BC, and AC together. Simple addition is all that is required.

So, let's do it:

Perimeter = AB + BC + AC Perimeter ≈ 15.23 + 10 + 7.21 Perimeter ≈ 32.44 units

And there you have it! The perimeter of triangle ABC is approximately 32.44 units. We're making great progress, guys! We've conquered the perimeter; now let's focus on the area. This is where things get even more interesting, because there are a couple of cool ways we can solve it.

Summing up the Perimeter

  • Side AB: Approximately 15.23 units
  • Side BC: 10 units
  • Side AC: Approximately 7.21 units
  • Perimeter: Approximately 32.44 units

With these calculations in hand, it's time to find the area. There are a couple of ways you can achieve this. You can either use the determinant formula, or you can use the more common formula of area = 1/2 * base * height, in this case it is much easier to use.

Determining the Area of Triangle ABC

Now, let's figure out the area of our triangle. There are a couple of ways to do this. We can use the formula that is derived from the Shoelace formula, but for a triangle like this, it is easier to think of the triangle in the Cartesian plane. Because of this, we can also use the area formula that you're probably most familiar with: Area = 1/2 * base * height. This is especially easy in this case because side BC lies on a horizontal line (y = 2). This means that we can directly find the height of the triangle from vertex A.

  1. Choosing the Base and Height:

    • We can use BC as the base of our triangle. We already know its length is 10 units.
    • The height is the perpendicular distance from vertex A to the line segment BC. Since BC is a horizontal line, the height is simply the difference in the y-coordinates of A and any point on BC. In this case, the height will be 8 - 2 = 6 units.

  2. Applying the Area Formula:

    • Area = 1/2 * base * height
    • Area = 1/2 * 10 * 6
    • Area = 1/2 * 60
    • Area = 30 square units

There you have it! The area of triangle ABC is 30 square units. Nice work, everyone! Now, to show you a second way to calculate the area, let's explore the shoelace method, which is pretty fun.

The Shoelace Method for Calculating the Area

Alright, let's have some fun with the Shoelace Method! This is a cool, visual way to calculate the area of a polygon, and it's particularly useful when you have the coordinates of the vertices. It's also known as the Gauss area formula, so you might run into that name as well.

  1. List the Coordinates:

    • Write down the coordinates of the vertices in a column, and then repeat the first vertex at the end to close the loop:
      • (2, 8)
      • (16, 2)
      • (6, 2)
      • (2, 8)

  2. Multiply Diagonally:

    • Multiply the x-coordinate of each point by the y-coordinate of the next point, and add the results:
      • (2 * 2) + (16 * 2) + (6 * 8) = 4 + 32 + 48 = 84

  3. Multiply the Other Way:

    • Multiply the x-coordinate of each point by the y-coordinate of the previous point, and add those results:
      • (16 * 8) + (6 * 2) + (2 * 2) = 128 + 12 + 4 = 144

  4. Subtract and Divide:

    • Subtract the sum from step 3 from the sum in step 2, and take the absolute value of the result. Then, divide by 2:
      • Area = |(84 - 144)| / 2
      • Area = |-60| / 2
      • Area = 60 / 2
      • Area = 30 square units

Summarizing the Area

  • Method 1: Using Base and Height, the area = 30 square units
  • Method 2: The Shoelace Method, the area = 30 square units

Conclusion: Wrapping Things Up

So, there you have it, guys! We've successfully calculated both the perimeter and the area of triangle ABC. We used the distance formula to find the side lengths, then summed them up to get the perimeter. For the area, we used the familiar formula and the Shoelace method as well. Geometry problems can seem daunting at first, but by breaking them down into smaller steps and using the right formulas, we can conquer them! Keep practicing, and you'll become a geometry whiz in no time. Thanks for following along! Keep learning and keep having fun with math! If you enjoyed this, check out other examples or problems online to keep practicing your geometry skills!