Calculating the Lengths of the Sides of Triangle DEF

Oct 13, 2025 by miodeya.com 53 views

Introduction

In geometry, a scalene triangle is a type of triangle that has all sides of different lengths. In other words, none of the sides of a scalene triangle are congruent. In this article, we will explore whether a given triangle DEF is a scalene triangle or not. To do this, we need to calculate the lengths of the sides of the triangle and then determine if any of them are congruent.

Calculating the Lengths of the Sides of Triangle DEF

To calculate the lengths of the sides of triangle DEF, we can use the distance formula. The distance formula is given by:

d = √((x2 - x1)² + (y2 - y1)²)

where (x1, y1) and (x2, y2) are the coordinates of two points.

Let's calculate the lengths of the sides of triangle DEF:

DE: The coordinates of points D and E are (-9, -3) and (-5, 6) respectively. Using the distance formula, we get: DE = √((-5 - (-9))² + (6 - (-3))²) = √((4)² + (9)²) = √(16 + 81) = √97
EF: The coordinates of points E and F are (-5, 6) and (4, 1) respectively. Using the distance formula, we get: EF = √((4 - (-5))² + (1 - 6)²) = √((9)² + (-5)²) = √(81 + 25) = √106
DF: The coordinates of points D and F are (-9, -3) and (4, 1) respectively. Using the distance formula, we get: DF = √((4 - (-9))² + (1 - (-3))²) = √((13)² + (4)²) = √(169 + 16) = √185

Determining if Triangle DEF is a Scalene Triangle

Now that we have calculated the lengths of the sides of triangle DEF, we can determine if it is a scalene triangle or not. A scalene triangle has all sides of different lengths. Let's compare the lengths of the sides of triangle DEF:

DE = √97
EF = √106
DF = √185

We can see that none of the sides of triangle DEF are congruent. The length of DE is √97, the length of EF is √106, and the length of DF is √185. Since none of the sides are congruent, we can conclude that triangle DEF is a scalene triangle.

Conclusion

In this article, we explored whether a given triangle DEF is a scalene triangle or not. We calculated the lengths of the sides of the triangle using the distance formula and then determined if any of them are congruent. Since none of the sides of triangle DEF are congruent, we can conclude that triangle DEF is a scalene triangle.

Justification

The correct answer is B. Yes, none of the sides are congruent. This is because none of the sides of triangle DEF are congruent, as we calculated the lengths of the sides and compared them.

Discussion

This problem is a great example of how to use the distance formula to calculate the lengths of the sides of a triangle. It also highlights the importance of comparing the lengths of the sides of a triangle to determine if it is a scalene triangle or not.

Key Takeaways

A scalene triangle is a type of triangle that has all sides of different lengths.
To determine if a triangle is a scalene triangle, we need to calculate the lengths of the sides of the triangle using the distance formula.
If none of the sides of a triangle are congruent, then the triangle is a scalene triangle.

References

Note: The references provided are for further reading and are not required to complete the problem.

Introduction

In our previous article, we explored the concept of scalene triangles and how to determine if a given triangle is a scalene triangle or not. In this article, we will provide a Q&A guide to help you better understand scalene triangles and how to work with them.

Q: What is a scalene triangle?

A: A scalene triangle is a type of triangle that has all sides of different lengths. In other words, none of the sides of a scalene triangle are congruent.

Q: How do I determine if a triangle is a scalene triangle?

A: To determine if a triangle is a scalene triangle, you need to calculate the lengths of the sides of the triangle using the distance formula. If none of the sides of the triangle are congruent, then the triangle is a scalene triangle.

Q: What is the distance formula?

A: The distance formula is a mathematical formula used to calculate the distance between two points in a coordinate plane. The formula is given by:

d = √((x2 - x1)² + (y2 - y1)²)

where (x1, y1) and (x2, y2) are the coordinates of two points.

Q: How do I use the distance formula to calculate the lengths of the sides of a triangle?

A: To use the distance formula to calculate the lengths of the sides of a triangle, you need to substitute the coordinates of the vertices of the triangle into the formula. For example, if you have a triangle with vertices at (x1, y1), (x2, y2), and (x3, y3), you can use the distance formula to calculate the lengths of the sides of the triangle as follows:

DE = √((x2 - x1)² + (y2 - y1)²)
EF = √((x3 - x2)² + (y3 - y2)²)
DF = √((x3 - x1)² + (y3 - y1)²)

Q: What is the difference between a scalene triangle and an isosceles triangle?

A: A scalene triangle is a type of triangle that has all sides of different lengths, while an isosceles triangle is a type of triangle that has two sides of equal length.

Q: Can a triangle have all sides of equal length?

A: Yes, a triangle can have all sides of equal length. This type of triangle is called an equilateral triangle.

Q: How do I determine if a triangle is an equilateral triangle?

A: To determine if a triangle is an equilateral triangle, you need to calculate the lengths of the sides of the triangle using the distance formula. If all the sides of the triangle are congruent, then the triangle is an equilateral triangle.

Q: What is the relationship between scalene triangles and equilateral triangles?

A: Scalene triangles and equilateral triangles are two types of triangles that are mutually exclusive. In other words, a triangle cannot be both a scalene triangle and an equilateral triangle at the same time.

Q: Can a triangle have two sides of equal length and one side of a different length?

A: Yes, a triangle can have two sides of equal length and one side of a different length. This type of triangle is called an isosceles triangle.

Q: How do I determine if a triangle is an isosceles triangle?

A: To determine if a triangle is an isosceles triangle, you need to calculate the lengths of the sides of the triangle using the distance formula. If two of the sides of the triangle are congruent, then the triangle is an isosceles triangle.

Conclusion

In this article, we provided a Q&A guide to help you better understand scalene triangles and how to work with them. We covered topics such as the definition of a scalene triangle, how to determine if a triangle is a scalene triangle, and the relationship between scalene triangles and other types of triangles.

Key Takeaways

A scalene triangle is a type of triangle that has all sides of different lengths.
To determine if a triangle is a scalene triangle, you need to calculate the lengths of the sides of the triangle using the distance formula.
A triangle can have all sides of equal length, which is called an equilateral triangle.
A triangle can have two sides of equal length and one side of a different length, which is called an isosceles triangle.