Triangle ABC Analysis: Acute, Equilateral, Or Scalene?

by TextBrain Team 55 views

Hey guys! Let's dive into a fun geometry problem. We've got a triangle, ABC, and the coordinates of its vertices: A(1, 2), B(5, -4), and C(-3, 2). Our mission? To figure out what kind of triangle we're dealing with. Is it acute, equilateral, or something else entirely? This is like a geometric detective story, and we're the investigators! We'll use some cool math tricks, like calculating distances and analyzing angles, to crack the case. Get ready to sharpen your pencils, because it's time to explore the wonderful world of triangles. Let's get started, shall we?

Determining the Triangle's Sides: Distance Formula to the Rescue

First things first, to understand our triangle, we need to know the lengths of its sides. We can't just eyeball it – we need to get precise. This is where the distance formula comes in handy. Remember that gem from your math classes? It's our secret weapon for finding the distance between two points on a coordinate plane. The distance formula is derived from the Pythagorean theorem, and it's a lifesaver in situations like this. Let's break down how we apply it to find the lengths of the sides of our triangle, namely AB, BC, and AC. Keep in mind that the distance formula works by essentially creating a right-angled triangle and applying Pythagoras to it.

Calculating AB

To find the length of side AB, we'll use the coordinates of points A(1, 2) and B(5, -4). The distance formula is: √[(x₂ - x₁)² + (y₂ - y₁)²]. Let's plug in the values: √[(5 - 1)² + (-4 - 2)²] = √[(4)² + (-6)²] = √(16 + 36) = √52. So, the length of AB is √52 units. That's a good start, right?

Calculating BC

Next up, we need to find the length of side BC. We'll use the coordinates of points B(5, -4) and C(-3, 2). Using the distance formula again: √[(-3 - 5)² + (2 - (-4))²] = √[(-8)² + (6)²] = √(64 + 36) = √100. This simplifies nicely to 10 units. BC is 10 units long. We're making progress, one side at a time!

Calculating AC

Finally, let's calculate the length of side AC. We'll use the coordinates of points A(1, 2) and C(-3, 2). Applying the distance formula: √[(-3 - 1)² + (2 - 2)²] = √[(-4)² + (0)²] = √(16 + 0) = √16. This simplifies to 4 units. AC is 4 units long. Now we have all three side lengths: AB = √52, BC = 10, and AC = 4. We're well on our way to understanding the type of triangle we have.

Unveiling the Triangle's Angles: The Cosine Rule

Now that we know the lengths of the sides, it's time to investigate the angles. To determine if the triangle is acute or not, we need to look at the angles. We can use the Law of Cosines to find the angles. The Law of Cosines is a powerful tool that relates the sides and angles of a triangle. It's especially useful when we know all the sides (which we do!). This law allows us to determine if any of the angles are obtuse or if all angles are acute. It's also a bit more involved than the distance formula, so let's take it slow and make sure we get it right. If any angle is greater than 90 degrees, the triangle is obtuse. If all angles are less than 90 degrees, the triangle is acute. If one angle is exactly 90 degrees, it's a right triangle.

Finding Angle A

Let's start by finding angle A. The Law of Cosines formula to find an angle is: cos(A) = (b² + c² - a²) / (2bc), where a is the side opposite angle A (BC), b is the side opposite angle B (AC), and c is the side opposite angle C (AB). Let's plug in our values: cos(A) = (4² + (√52)² - 10²) / (2 * 4 * √52) = (16 + 52 - 100) / (8√52) = -32 / (8√52). To find the angle A, we take the inverse cosine (arccos) of this value. Use a calculator to find arccos(-32 / (8√52)). This gives us an angle of approximately 110.5 degrees. Oh no! This means that our triangle has an obtuse angle.

The Verdict

Since we have found an angle greater than 90 degrees, we can immediately conclude that the triangle is not acute. The triangle is also not equilateral because the sides have different lengths. Therefore, it must be a scalene and obtuse triangle.

Classification: Acute, Obtuse, or Right?

So, after all this calculation, what kind of triangle have we got? Let's recap what we've found and classify the triangle properly. We know the lengths of the sides are: AB = √52, BC = 10, and AC = 4. We calculated the angles, and we found that at least one angle is greater than 90 degrees.

  • Acute Triangle: All angles are less than 90 degrees.
  • Obtuse Triangle: One angle is greater than 90 degrees.
  • Right Triangle: One angle is exactly 90 degrees.

Because we determined that one angle (angle A) is approximately 110.5 degrees, our triangle is an obtuse triangle. It's not acute, nor is it a right triangle. Moreover, since all the sides have different lengths, it's a scalene triangle as well. So, the final answer is: Triangle ABC is a scalene, obtuse triangle.

Conclusion: The Mystery Solved!

And there you have it, guys! We've successfully analyzed triangle ABC. We've used the distance formula to find the sides, the Law of Cosines to find the angles, and we've determined that the triangle is scalene and obtuse. Math can be fun, right? Keep practicing, and you'll become geometry wizards in no time! Remember, the key is to understand the formulas, apply them correctly, and break down the problem step by step. Great job, everyone!