Problem 20 - Prealgebra

Question

GEOMETRY Is \(\triangle A B C\) with vertices \(A(8,4), B(-2,7),\) and \(C(0,9)\) a scalene triangle? Explain.

Step-by-Step Solution

Answer

Yes, \(\triangle ABC\) is a scalene triangle because all three sides have different lengths.

Step 1: Understand the Problem

To determine if \(\triangle ABC\) is a scalene triangle, we need to verify if all three sides of the triangle have different lengths. A scalene triangle is one in which no two sides are of equal length.

Step 2: Use the Distance Formula

The distance formula is \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). We will use this formula to find the lengths of the sides \(AB\), \(BC\), and \(CA\).

Step 3: Calculate Length of AB

Using the distance formula, we calculate the distance between points \(A(8,4)\) and \(B(-2,7)\): \[d_{AB} = \sqrt{(-2 - 8)^2 + (7 - 4)^2} = \sqrt{(-10)^2 + (3)^2} = \sqrt{100 + 9} = \sqrt{109}\]

Step 4: Calculate Length of BC

Now, find the distance between points \(B(-2,7)\) and \(C(0,9)\): \[d_{BC} = \sqrt{(0 + 2)^2 + (9 - 7)^2} = \sqrt{(2)^2 + (2)^2} = \sqrt{4 + 4} = \sqrt{8}\]

Step 5: Calculate Length of CA

Lastly, compute the distance between points \(C(0,9)\) and \(A(8,4)\): \[d_{CA} = \sqrt{(8 - 0)^2 + (4 - 9)^2} = \sqrt{(8)^2 + (-5)^2} = \sqrt{64 + 25} = \sqrt{89}\]

Step 6: Compare the Lengths of the Sides

The lengths of the sides are \(\sqrt{109}\), \(\sqrt{8}\), and \(\sqrt{89}\). Since \(\sqrt{109} eq \sqrt{8} eq \sqrt{89}\), all three sides have different lengths.

Step 7: Conclusion

Since all side lengths are different, \(\triangle ABC\) is a scalene triangle.

Key Concepts

Distance Formula

The distance formula is fundamental in coordinate geometry and is used to calculate the distance between two points in a plane. It is derived from the Pythagorean Theorem, and it is given by \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]where

• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.
• \((x_2 - x_1)^2\) represents the square of the difference in the x-coordinates.
• \((y_2 - y_1)^2\) represents the square of the difference in the y-coordinates.

This formula helps to determine the actual linear distance between two points and is crucial for solving various problems in geometry. For instance, in the problem of determining whether a triangle is scalene, the distance formula is employed to find the lengths of the sides of the triangle.
This allows for the comparison of these lengths to classify the triangle correctly.

Geometry Problem Solving

Solving problems in geometry often involves a systematic approach to ensure accuracy and efficiency. Begin by identifying what the problem is asking. In the case of classifying a triangle, it’s essential to first understand the characteristics of the different types of triangles.
Once you know what to look for, utilize the appropriate mathematical formulas and techniques. Using the given coordinates, apply the distance formula to find side lengths, which will be necessary to classify the triangle.
Break down the problem into smaller steps:

• Understanding the problem: Here, you need to determine whether all sides have different lengths to identify a scalene triangle.
• Calculating side lengths: Apply the distance formula to each pair of points that represent the triangle's vertices.
• Comparing results: Assess the calculated lengths to check if all sides are of different dimensions.

Being methodical in your approach helps avoid errors and improves problem-solving skills in geometry.

Triangle Classification

Triangles are classified based on side lengths and angle measurements. The focus here is side-based classification which includes scalene, isosceles, and equilateral triangles:

• **Scalene Triangle:** All sides are of different lengths. None of the angles are equal.
• **Isosceles Triangle:** At least two sides are equal in length, and at least two angles are equal.
• **Equilateral Triangle:** All sides are of equal length, and all angles are equal (each measuring 60 degrees).

Identifying the type of triangle involves verifying these properties. For a scalene triangle, as demonstrated in the solution, all three sides must be different in length. By applying the distance formula to find the lengths of the sides, one can determine whether no sides are equal. This exercise reinforces an understanding of triangle properties while honing skills in calculating and comparing measurements in geometric figures.