Classifying Triangle With Vertices (-3,-5), (7,4), And (0,4)
Determining the nature of a triangle given its vertices is a fundamental problem in coordinate geometry. In this comprehensive analysis, we will delve into the characteristics of the triangle formed by the points (-3,-5), (7,4), and (0,4). Our exploration will involve calculating side lengths using the distance formula, analyzing slopes to identify perpendicular sides, and ultimately classifying the triangle based on its properties. By meticulously examining these geometric attributes, we can accurately determine the type of triangle represented by these vertices.
Calculating Side Lengths
The foundation of understanding a triangle lies in its side lengths. To determine these, we employ the distance formula, a cornerstone of coordinate geometry. The distance formula elegantly calculates the distance between two points in a coordinate plane, considering both the horizontal and vertical separation. Let's denote our vertices as A(-3, -5), B(7, 4), and C(0, 4). This notation will streamline our calculations and discussions.
The distance formula is given by:
d = √[(x₂ - x₁)² + (y₂ - y₁)²]
Where (x₁, y₁) and (x₂, y₂) represent the coordinates of the two points.
Side AB
Let's first calculate the length of side AB, which connects points A(-3, -5) and B(7, 4). Applying the distance formula:
AB = √[(7 - (-3))² + (4 - (-5))²] = √[(10)² + (9)²] = √(100 + 81) = √181
Therefore, the length of side AB is √181 units.
Side BC
Next, we calculate the length of side BC, connecting points B(7, 4) and C(0, 4). Using the distance formula:
BC = √[(0 - 7)² + (4 - 4)²] = √[(-7)² + (0)²] = √49 = 7
Thus, the length of side BC is 7 units.
Side AC
Finally, we calculate the length of side AC, connecting points A(-3, -5) and C(0, 4). Applying the distance formula:
AC = √[(0 - (-3))² + (4 - (-5))²] = √[(3)² + (9)²] = √(9 + 81) = √90 = 3√10
Hence, the length of side AC is 3√10 units.
Now that we have the lengths of all three sides – AB = √181, BC = 7, and AC = 3√10 – we can proceed to analyze these lengths to classify the triangle. The side lengths provide crucial information about whether the triangle is scalene (all sides different), isosceles (two sides equal), or equilateral (all sides equal).
Analyzing Slopes to Identify Perpendicular Sides
To further classify the triangle, particularly to determine if it is a right triangle, we need to analyze the slopes of its sides. The slope of a line segment provides information about its direction and steepness. Two lines are perpendicular if and only if the product of their slopes is -1. This property is a cornerstone of coordinate geometry and is vital for identifying right angles in geometric figures.
The slope formula is given by:
m = (y₂ - y₁) / (x₂ - x₁)
Where (x₁, y₁) and (x₂, y₂) are the coordinates of two points on the line.
Slope of AB
Let's calculate the slope of side AB, connecting points A(-3, -5) and B(7, 4). Applying the slope formula:
mAB = (4 - (-5)) / (7 - (-3)) = 9 / 10
Therefore, the slope of side AB is 9/10.
Slope of BC
Next, we calculate the slope of side BC, connecting points B(7, 4) and C(0, 4). Using the slope formula:
mBC = (4 - 4) / (0 - 7) = 0 / -7 = 0
Thus, the slope of side BC is 0. This indicates that side BC is a horizontal line.
Slope of AC
Finally, we calculate the slope of side AC, connecting points A(-3, -5) and C(0, 4). Applying the slope formula:
mAC = (4 - (-5)) / (0 - (-3)) = 9 / 3 = 3
Hence, the slope of side AC is 3.
Now that we have the slopes of all three sides – mAB = 9/10, mBC = 0, and mAC = 3 – we can check for perpendicularity. To do this, we look for pairs of slopes whose product is -1. In this case, we need to examine the products of the slopes of AB and BC, AB and AC, and BC and AC.
Checking for Perpendicularity
Let's analyze the products of the slopes:
- mAB * mBC = (9/10) * 0 = 0 ≠ -1
- mAB * mAC = (9/10) * 3 = 27/10 ≠ -1
- mBC * mAC = 0 * 3 = 0 ≠ -1
Since none of the products of the slopes equal -1, we can conclude that none of the sides are perpendicular to each other. This means that the triangle does not have a right angle and is therefore not a right triangle.
Classifying the Triangle
Having calculated the side lengths and analyzed the slopes, we are now equipped to classify the triangle formed by the vertices (-3,-5), (7,4), and (0,4). Our classification will be based on the lengths of the sides and the presence or absence of right angles. We have determined the following:
- Side lengths: AB = √181, BC = 7, AC = 3√10
- Slopes: mAB = 9/10, mBC = 0, mAC = 3
- No right angles (since no pair of sides has slopes whose product is -1)
Based on Side Lengths
Let's compare the side lengths:
- AB = √181 ≈ 13.45
- BC = 7
- AC = 3√10 ≈ 9.49
Since all three side lengths are different (√181 ≠ 7 ≠ 3√10), the triangle is scalene. A scalene triangle is defined as a triangle in which all three sides have different lengths.
Based on Angles
We have already established that the triangle does not have a right angle. Therefore, it is not a right triangle. Additionally, since the slopes of the sides are different, none of the angles are equal. This further supports the conclusion that the triangle is scalene and not a right triangle.
Final Classification
Combining our analysis of side lengths and angles, we can definitively classify the triangle. The triangle with vertices (-3, -5), (7, 4), and (0, 4) is a scalene triangle. This classification is based on the fact that all three sides have different lengths and there are no right angles within the triangle. The unique combination of side lengths and angles provides a clear geometric signature for this particular triangle.
Summary of Findings
In this comprehensive analysis, we have meticulously examined the triangle formed by the vertices (-3,-5), (7,4), and (0,4). Our approach involved calculating side lengths using the distance formula, analyzing slopes to identify perpendicular sides, and synthesizing this information to classify the triangle. The key steps and findings are summarized below:
- Side Lengths Calculation: We employed the distance formula to determine the lengths of the sides:
- AB = √181
- BC = 7
- AC = 3√10
- Slope Analysis: We calculated the slopes of the sides to identify any perpendicular relationships:
- mAB = 9/10
- mBC = 0
- mAC = 3
- Perpendicularity Check: We verified that no pair of sides has slopes whose product is -1, indicating that the triangle does not have a right angle.
- Classification Based on Side Lengths: Since all three sides have different lengths (√181 ≠ 7 ≠ 3√10), we classified the triangle as scalene.
- Classification Based on Angles: The absence of right angles further confirms that the triangle is not a right triangle.
- Final Classification: The triangle with vertices (-3, -5), (7, 4), and (0, 4) is a scalene triangle.
This classification provides a complete geometric description of the triangle, highlighting its unique properties and characteristics. The systematic approach used in this analysis underscores the power of coordinate geometry in dissecting and understanding geometric figures. By applying fundamental formulas and principles, we can accurately classify triangles and other geometric shapes based on their vertices and coordinates.
Conclusion
In conclusion, the meticulous analysis of the triangle with vertices (-3,-5), (7,4), and (0,4) has unequivocally demonstrated that it is a scalene triangle. This classification is rooted in the precise calculations of side lengths and the examination of slopes, which collectively revealed the absence of equal sides and right angles. The journey through this geometric exploration has not only identified the triangle's type but also underscored the elegance and utility of coordinate geometry in unraveling the properties of shapes.
By applying the distance formula and the slope formula, we were able to quantify the essential attributes of the triangle, thereby facilitating its classification. The methodical calculation of side lengths allowed us to discern that all three sides were of different lengths, a defining characteristic of a scalene triangle. Furthermore, the analysis of slopes confirmed the absence of perpendicular sides, thus ruling out the possibility of it being a right triangle.
This exercise serves as a testament to the power of mathematical tools in dissecting and understanding geometric concepts. The principles of coordinate geometry provide a robust framework for analyzing shapes, determining their properties, and ultimately classifying them with precision. As we conclude this detailed examination, we are reminded of the intrinsic beauty and order inherent in the world of mathematics, where every point and every line holds a story waiting to be uncovered.
