Coordinates A(-3,-3), B(-2,2), C(5,6): Math Exploration

Hey guys! Today, we're diving deep into the world of coordinate geometry with a specific set of points: A(-3,-3), B(-2,2), and C(5,6). These seemingly simple coordinates open up a fascinating array of mathematical investigations. We can explore distances, slopes, areas, and even the types of geometric figures these points might form. Let's break it down step-by-step and see what we can uncover together.

1. Understanding the Basics: Plotting and Visualizing the Points

Before we jump into calculations, it’s crucial to visualize what we're working with. Think of a standard Cartesian plane, the one with the x-axis running horizontally and the y-axis running vertically. Plotting these points helps us get a feel for their relative positions and potential relationships.

• Point A (-3, -3): This point is located in the third quadrant, where both x and y values are negative. Imagine moving 3 units to the left along the x-axis and then 3 units down along the y-axis. That's where A sits.
• Point B (-2, 2): Point B resides in the second quadrant, where x is negative, and y is positive. From the origin, we move 2 units left and then 2 units up.
• Point C (5, 6): Finally, Point C is in the first quadrant, where both x and y are positive. We move 5 units to the right and then a whopping 6 units up. See how visualizing makes it easier to grasp?

By plotting these points, we can start to speculate what kind of shape they might form. Could they be the vertices of a triangle? Maybe even a special type of triangle, like a right triangle or an isosceles triangle? This initial visualization is our first step in this mathematical journey. Remember, coordinate geometry is all about bridging the gap between algebra and geometry, and visualization is key.

2. Calculating Distances: The Distance Formula

Now that we've visualized our points, let's get down to some calculations! The first thing that often comes to mind when dealing with coordinates is finding the distances between them. This is where the distance formula comes in handy. It's a direct application of the Pythagorean theorem and allows us to precisely calculate the length of a line segment connecting any two points.

The distance formula is:

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

Where:

• d is the distance between the two points
• (x₁, y₁) are the coordinates of the first point
• (x₂, y₂) are the coordinates of the second point

Let's apply this formula to our points:

Distance Between A and B (AB)

• A (-3, -3) as (x₁, y₁) and B (-2, 2) as (x₂, y₂)
• AB = √[(-2 - (-3))² + (2 - (-3))²]
• AB = √[(1)² + (5)²]
• AB = √(1 + 25)
• AB = √26

Distance Between B and C (BC)

• B (-2, 2) as (x₁, y₁) and C (5, 6) as (x₂, y₂)
• BC = √[(5 - (-2))² + (6 - 2)²]
• BC = √[(7)² + (4)²]
• BC = √(49 + 16)
• BC = √65

Distance Between A and C (AC)

• A (-3, -3) as (x₁, y₁) and C (5, 6) as (x₂, y₂)
• AC = √[(5 - (-3))² + (6 - (-3))²]
• AC = √[(8)² + (9)²]
• AC = √(64 + 81)
• AC = √145

So, we've calculated the lengths of the sides of our potential triangle: AB = √26, BC = √65, and AC = √145. These distances are crucial for determining what type of triangle, if any, these points form. Next up, we'll see how these distances help us classify the triangle. This step really shows how vital the distance formula is in coordinate geometry. It's the foundation for many other calculations and analyses.

3. Determining Triangle Type: Scalene, Isosceles, or Right?

With the distances between our points calculated, we're now equipped to classify the triangle formed by A(-3,-3), B(-2,2), and C(5,6). Remember, we have three main categories to consider: scalene, isosceles, and right triangles. Each has a unique set of properties based on side lengths and angles. The distances we calculated—AB = √26, BC = √65, and AC = √145—are our key to unlocking the triangle's identity.

Scalene Triangle

A scalene triangle is the most general type, characterized by having all three sides of different lengths. Looking at our calculated distances, √26, √65, and √145 are indeed all distinct values. This is a strong indication that we might be dealing with a scalene triangle. However, we can’t definitively say it's scalene until we’ve ruled out other possibilities, particularly the right triangle.

Isosceles Triangle

An isosceles triangle has at least two sides of equal length. In our case, none of the side lengths (√26, √65, √145) are equal. So, we can confidently say that our triangle is not isosceles.

Right Triangle

A right triangle is defined by having one angle that measures 90 degrees. The hallmark of a right triangle is that its sides satisfy the Pythagorean theorem: a² + b² = c², where c is the longest side (the hypotenuse). Let’s see if our side lengths fit this pattern:

• a² = (√26)² = 26
• b² = (√65)² = 65
• c² = (√145)² = 145

Now, let’s check if a² + b² = c²:

26 + 65 = 91

91 ≠ 145

Since the Pythagorean theorem doesn't hold, we can conclude that our triangle is not a right triangle. This is a crucial step in our triangle classification process. Using the Pythagorean theorem is a powerful tool for identifying right triangles in coordinate geometry.

Conclusion: The Scalene Verdict

Having eliminated the possibilities of isosceles and right triangles, we can confidently conclude that triangle ABC is a scalene triangle. This means all three of its sides have different lengths, and none of its angles are right angles. See how we used a process of elimination combined with the properties of different triangles to reach our conclusion?

4. Calculating Slopes: Determining Parallel and Perpendicular Lines

Beyond distances and triangle types, the slopes of the lines connecting our points offer another layer of insight. The slope of a line tells us how steep it is and in what direction it rises or falls. Calculating slopes is essential for determining if lines are parallel, perpendicular, or neither. The slope formula is a fundamental tool in coordinate geometry, and let's put it to work with our points A(-3,-3), B(-2,2), and C(5,6).

The slope (m) between two points (x₁, y₁) and (x₂, y₂) is calculated as:

m = (y₂ - y₁) / (x₂ - x₁)

Let's find the slopes of the lines formed by our points:

Slope of Line AB (mAB)

• A (-3, -3) as (x₁, y₁) and B (-2, 2) as (x₂, y₂)
• mAB = (2 - (-3)) / (-2 - (-3))
• mAB = (5) / (1)
• mAB = 5

Slope of Line BC (mBC)

• B (-2, 2) as (x₁, y₁) and C (5, 6) as (x₂, y₂)
• mBC = (6 - 2) / (5 - (-2))
• mBC = (4) / (7)
• mBC = 4/7

Slope of Line AC (mAC)

• A (-3, -3) as (x₁, y₁) and C (5, 6) as (x₂, y₂)
• mAC = (6 - (-3)) / (5 - (-3))
• mAC = (9) / (8)
• mAC = 9/8

Now that we have the slopes—mAB = 5, mBC = 4/7, and mAC = 9/8—we can analyze the relationships between the lines.

Parallel Lines

Parallel lines have the same slope. Looking at our slopes, none of them are equal, so no pair of lines formed by these points are parallel. Understanding this concept of parallel lines and equal slopes is a cornerstone of coordinate geometry.

Perpendicular Lines

Perpendicular lines have slopes that are negative reciprocals of each other. This means if one line has a slope of m, a line perpendicular to it will have a slope of -1/m. Let's check our slopes:

• Is mAB perpendicular to mBC? 5 and 4/7 are not negative reciprocals.
• Is mAB perpendicular to mAC? 5 and 9/8 are not negative reciprocals.
• Is mBC perpendicular to mAC? 4/7 and 9/8 are not negative reciprocals.

Therefore, no pair of lines formed by these points are perpendicular. This analysis of perpendicular lines and their slopes further reinforces our understanding of the triangle's properties.

Conclusion: No Parallel or Perpendicular Lines

Based on our slope calculations, we've determined that the lines formed by points A, B, and C are neither parallel nor perpendicular. This aligns with our earlier conclusion that the triangle is scalene, as right triangles would necessarily have perpendicular sides. Analyzing slopes provides a deeper understanding of the geometric relationships between the points. It's a powerful technique in coordinate geometry that allows us to explore the orientation and spatial arrangement of lines.

5. Calculating the Area: Using the Shoelace Formula

So, we've classified the triangle, analyzed its sides and slopes, but what about its area? Calculating the area is another fundamental aspect of coordinate geometry, and for triangles defined by coordinates, the Shoelace Formula (also known as the Gauss's area formula) is a particularly elegant and efficient method. This formula bypasses the need to find a base and height, instead relying directly on the coordinates of the vertices.

The Shoelace Formula for a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) is:

Area = 0.5 * |(x₁y₂ + x₂y₃ + x₃y₁) - (y₁x₂ + y₂x₃ + y₃x₁)|

Let's apply this formula to our points A(-3,-3), B(-2,2), and C(5,6):

• (x₁, y₁) = (-3, -3)
• (x₂, y₂) = (-2, 2)
• (x₃, y₃) = (5, 6)

Now, plug the values into the formula:

Area = 0.5 * |((-3)(2) + (-2)(6) + (5)(-3)) - ((-3)(-2) + (2)(5) + (6)(-3))|

Area = 0.5 * |(-6 - 12 - 15) - (6 + 10 - 18)|

Area = 0.5 * |(-33) - (-2)|

Area = 0.5 * |-33 + 2|

Area = 0.5 * |-31|

Area = 0.5 * 31

Area = 15.5 square units

Wow! We've calculated the area of our scalene triangle ABC to be 15.5 square units. The Shoelace Formula is a powerful tool, especially when dealing with coordinates directly. It eliminates the need to calculate heights or angles, providing a straightforward path to the area. This demonstrates the interconnectedness of coordinate geometry, where different formulas and concepts work together to reveal various properties of geometric figures.

Conclusion: A Comprehensive Exploration of Coordinates

So, guys, we've journeyed through a comprehensive exploration of the coordinates A(-3,-3), B(-2,2), and C(5,6). We started by visualizing the points, then calculated distances to classify the triangle as scalene. We delved into slopes to confirm the absence of parallel or perpendicular lines and finally computed the area using the Shoelace Formula.

This exercise highlights the power and versatility of coordinate geometry. By applying a combination of formulas and concepts, we can unravel the geometric properties of shapes defined by coordinates. Remember, mathematics isn't just about memorizing formulas; it's about understanding how they connect and using them to solve problems! Keep exploring, keep questioning, and most importantly, keep having fun with math!