Determining the nature of a triangle given its vertices is a fundamental problem in coordinate geometry. This article delves into the specifics of the triangle formed by the points (10,7), (6,2), and (3,2). By calculating the lengths of the sides and analyzing their relationships, we can classify the triangle as scalene, isosceles, equilateral, or right-angled. This exploration involves the application of the distance formula and the Pythagorean theorem, providing a comprehensive understanding of triangle properties and their identification in the coordinate plane.
Determining Triangle Type from Vertices
To determine what can be said of the triangle with vertices (10,7), (6,2), and (3,2), we need to analyze the lengths of its sides. This analysis will help us classify the triangle based on its side lengths and angles. The primary classifications based on side lengths are scalene, isosceles, and equilateral, while considering angles, we can identify if the triangle is a right-angled triangle. The approach involves using the distance formula to calculate the lengths of the sides and then applying the Pythagorean theorem to check for a right angle. Understanding these fundamental concepts is crucial in coordinate geometry and helps in solving a variety of geometric problems.
Calculating Side Lengths
To begin, we will use the distance formula to calculate the lengths of the sides of the triangle. The distance formula, derived from the Pythagorean theorem, is given by:
√((x₂ - x₁)² + (y₂ - y₁)²)
where (x₁, y₁) and (x₂, y₂) are the coordinates of the two points. Let's denote the vertices as A(10,7), B(6,2), and C(3,2). We will calculate the lengths of sides AB, BC, and CA.
-
Length of AB: Using the distance formula for points A(10,7) and B(6,2): AB = √((6 - 10)² + (2 - 7)²) = √((-4)² + (-5)²) = √(16 + 25) = √41
-
Length of BC: Using the distance formula for points B(6,2) and C(3,2): BC = √((3 - 6)² + (2 - 2)²) = √((-3)² + (0)²) = √(9 + 0) = √9 = 3
-
Length of CA: Using the distance formula for points C(3,2) and A(10,7): CA = √((10 - 3)² + (7 - 2)²) = √((7)² + (5)²) = √(49 + 25) = √74
We have now determined the lengths of the sides of the triangle: AB = √41, BC = 3, and CA = √74. These lengths are crucial for classifying the triangle based on its properties.
Classifying by Side Lengths: Scalene, Isosceles, or Equilateral
Now that we have the lengths of the sides, we can classify the triangle based on these lengths. Recall the definitions:
- Scalene Triangle: A triangle with all three sides of different lengths.
- Isosceles Triangle: A triangle with at least two sides of equal length.
- Equilateral Triangle: A triangle with all three sides of equal length.
From our calculations, the side lengths are AB = √41, BC = 3, and CA = √74. Since all three sides have different lengths (√41 ≈ 6.40, 3, and √74 ≈ 8.60), the triangle is a scalene triangle. This classification is based purely on the side lengths, and it tells us that no two sides of this triangle are equal.
Checking for a Right Angle: Applying the Pythagorean Theorem
To determine if the triangle is a right-angled triangle, we can apply the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, this is expressed as:
a² + b² = c²
where c is the length of the hypotenuse, and a and b are the lengths of the other two sides. In our case, the sides are AB = √41, BC = 3, and CA = √74. We need to check if the sum of the squares of the two shorter sides equals the square of the longest side.
Let's check if BC² + AB² = CA²:
3² + (√41)² = 9 + 41 = 50
(√74)² = 74
Since 50 ≠ 74, the Pythagorean theorem does not hold true for this triangle. Therefore, the triangle is not a right-angled triangle. This means that none of the angles in the triangle is a right angle (90 degrees).
Conclusion: Triangle Classification
In conclusion, by analyzing the side lengths and applying the Pythagorean theorem, we have classified the triangle with vertices (10,7), (6,2), and (3,2). The side lengths were calculated as AB = √41, BC = 3, and CA = √74. Based on these lengths, the triangle is a scalene triangle because all three sides have different lengths. Additionally, by applying the Pythagorean theorem, we determined that the triangle is not a right-angled triangle. Therefore, the most accurate description of this triangle is that it is a scalene triangle.
This exercise demonstrates the importance of understanding and applying geometric principles like the distance formula and the Pythagorean theorem in coordinate geometry. Classifying triangles based on their properties is a fundamental skill in mathematics and has applications in various fields, including engineering, physics, and computer graphics.
By meticulously analyzing the given vertices and applying the relevant formulas, we can confidently classify the triangle and gain a deeper understanding of its characteristics. This approach not only answers the specific question but also reinforces the broader concepts of triangle geometry and coordinate systems.
Repair Input Keyword: What are the properties of the triangle formed by the vertices (10,7), (6,2), and (3,2)?
SEO Title: Triangle Properties Analysis Points (10,7), (6,2), (3,2) ️📐