What Type Of Triangle Is Formed By Joining The Points \[$ D(7,3) \$\], \[$ E(8,1) \$\], And \[$ F(4,-1) \$\]?A. Equilateral Triangle B. Isosceles Triangle C. Right Triangle D. Acute Scalene Triangle E. Obtuse Scalene Triangle

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What Type of Triangle is Formed by Joining the Points D(7,3), E(8,1), and F(4,-1)?

In geometry, triangles are classified based on their side lengths and angles. To determine the type of triangle formed by joining the points D(7,3), E(8,1), and F(4,-1), we need to calculate the lengths of the sides and the measures of the angles. In this article, we will explore the properties of each type of triangle and determine which one is formed by the given points.

To calculate the side lengths of the triangle, we can use the distance formula:

a = √((x2 - x1)^2 + (y2 - y1)^2)

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

Let's calculate the side lengths of the triangle:

  • DE: √((8 - 7)^2 + (1 - 3)^2) = √(1^2 + (-2)^2) = √(5)
  • EF: √((4 - 8)^2 + (-1 - 1)^2) = √((-4)^2 + (-2)^2) = √(20)
  • FD: √((4 - 7)^2 + (-1 - 3)^2) = √((-3)^2 + (-4)^2) = √(25)

Now that we have the side lengths, we can determine the type of triangle.

  • Equilateral Triangle: An equilateral triangle has all sides of equal length. Since the side lengths of the triangle are √5, √20, and √25, it is not an equilateral triangle.
  • Isosceles Triangle: An isosceles triangle has two sides of equal length. Since the side lengths of the triangle are √5, √20, and √25, it is not an isosceles triangle.
  • Right Triangle: A right triangle has one right angle (90 degrees). To determine if the triangle is a right triangle, we can use the Pythagorean theorem: a^2 + b^2 = c^2, where a and b are the legs of the triangle and c is the hypotenuse. Let's check if the triangle is a right triangle:
    • DE^2 + EF^2 = (√5)^2 + (√20)^2 = 5 + 20 = 25
    • (√25)^2 = 25 Since DE^2 + EF^2 = (√25)^2, the triangle is a right triangle.
  • Acute Scalene Triangle: An acute scalene triangle has all sides of different lengths and all angles less than 90 degrees. Since the triangle is a right triangle, it is not an acute scalene triangle.
  • Obtuse Scalene Triangle: An obtuse scalene triangle has all sides of different lengths and one angle greater than 90 degrees. Since the triangle is a right triangle, it is not an obtuse scalene triangle.

Based on the calculations, the triangle formed by joining the points D(7,3), E(8,1), and F(4,-1) is a Right Triangle.

Right triangles have several important properties:

  • Pythagorean Theorem: a^2 + b^2 = c^2, where a and b are the legs of the triangle and c is the hypotenuse.
  • Perpendicular Bisectors: The perpendicular bisectors of the legs of a right triangle intersect at the midpoint of the hypotenuse.
  • Altitude: The altitude of a right triangle is the line segment from the right angle to the hypotenuse.

Right triangles have many real-world applications:

  • Building Design: Right triangles are used in building design to create stable and balanced structures.
  • Physics: Right triangles are used in physics to calculate distances and velocities.
  • Engineering: Right triangles are used in engineering to design and build machines and mechanisms.

In conclusion, the triangle formed by joining the points D(7,3), E(8,1), and F(4,-1) is a Right Triangle. Right triangles have several important properties and have many real-world applications.
Frequently Asked Questions (FAQs) about Right Triangles

Right triangles are a fundamental concept in geometry and have many real-world applications. In this article, we will answer some frequently asked questions about right triangles.

Q: What is a right triangle?

A: A right triangle is a triangle with one right angle (90 degrees). The right angle is the angle between the two legs of the triangle.

Q: How do you identify a right triangle?

A: To identify a right triangle, you can use the Pythagorean theorem: a^2 + b^2 = c^2, where a and b are the legs of the triangle and c is the hypotenuse. If the sum of the squares of the legs is equal to the square of the hypotenuse, then the triangle is a right triangle.

Q: What are the properties of a right triangle?

A: The properties of a right triangle include:

  • Pythagorean Theorem: a^2 + b^2 = c^2, where a and b are the legs of the triangle and c is the hypotenuse.
  • Perpendicular Bisectors: The perpendicular bisectors of the legs of a right triangle intersect at the midpoint of the hypotenuse.
  • Altitude: The altitude of a right triangle is the line segment from the right angle to the hypotenuse.

Q: What are the real-world applications of right triangles?

A: Right triangles have many real-world applications, including:

  • Building Design: Right triangles are used in building design to create stable and balanced structures.
  • Physics: Right triangles are used in physics to calculate distances and velocities.
  • Engineering: Right triangles are used in engineering to design and build machines and mechanisms.

Q: How do you calculate the area of a right triangle?

A: To calculate the area of a right triangle, you can use the formula: Area = (1/2) × base × height, where the base and height are the two legs of the triangle.

Q: How do you calculate the perimeter of a right triangle?

A: To calculate the perimeter of a right triangle, you can add the lengths of all three sides: Perimeter = a + b + c, where a and b are the legs of the triangle and c is the hypotenuse.

Q: What are the different types of right triangles?

A: There are several types of right triangles, including:

  • Isosceles Right Triangle: An isosceles right triangle has two legs of equal length.
  • Scalene Right Triangle: A scalene right triangle has all three sides of different lengths.
  • Obtuse Right Triangle: An obtuse right triangle has one angle greater than 90 degrees.

In conclusion, right triangles are a fundamental concept in geometry and have many real-world applications. By understanding the properties and types of right triangles, you can solve problems and make calculations with ease.

For more information on right triangles, you can check out the following resources:

  • Geometry Textbooks: Many geometry textbooks cover the topic of right triangles in detail.
  • Online Resources: Websites such as Khan Academy and Mathway offer interactive lessons and exercises on right triangles.
  • Math Software: Software such as GeoGebra and Mathematica can be used to visualize and explore right triangles.

Right triangles are an essential part of geometry and have many real-world applications. By understanding the properties and types of right triangles, you can solve problems and make calculations with ease. Whether you're a student or a professional, right triangles are an important concept to master.