Lydia Graphed \[$\triangle DEF\$\] At The Coordinates \[$D (-2, -1)\$\], \[$E (-2, 2)\$\], And \[$F (0, 0)\$\]. She Thinks \[$\triangle DEF\$\] Is A Right Triangle. Is Lydia's Assertion Correct?A. Yes; The Slopes

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Is Lydia's Assertion Correct? A Mathematical Analysis of Triangle DEF

In mathematics, a right triangle is a triangle with one right angle (90 degrees). To determine if a triangle is a right triangle, we can use various methods, including the Pythagorean theorem, slope calculations, and coordinate geometry. In this article, we will analyze the triangle DEF, graphed by Lydia at the coordinates D (-2, -1), E (-2, 2), and F (0, 0), to determine if it is a right triangle.

To begin, let's understand the coordinates of the triangle DEF. The coordinates of a point in a 2D plane are represented as (x, y), where x is the horizontal distance from the origin (0, 0) and y is the vertical distance from the origin. In this case, the coordinates of the triangle DEF are:

  • D (-2, -1)
  • E (-2, 2)
  • F (0, 0)

To determine if the triangle DEF is a right triangle, we can calculate the slopes of the lines formed by the points D, E, and F. The slope of a line is a measure of how steep it is and can be calculated using the formula:

m = (y2 - y1) / (x2 - x1)

where m is the slope, and (x1, y1) and (x2, y2) are the coordinates of two points on the line.

Let's calculate the slopes of the lines formed by the points D, E, and F:

  • Slope of DE: m = (2 - (-1)) / (-2 - (-2)) = 3 / 0 = undefined (vertical line)
  • Slope of EF: m = (0 - 2) / (0 - (-2)) = -2 / 2 = -1
  • Slope of FD: m = (-1 - 0) / (-2 - 0) = -1 / -2 = 1/2

Now that we have calculated the slopes of the lines formed by the points D, E, and F, let's analyze them to determine if the triangle DEF is a right triangle.

  • The slope of DE is undefined, which means that the line DE is vertical.
  • The slope of EF is -1, which means that the line EF has a negative slope.
  • The slope of FD is 1/2, which means that the line FD has a positive slope.

Based on the analysis of the slopes of the lines formed by the points D, E, and F, we can conclude that the triangle DEF is not a right triangle. A right triangle has one right angle (90 degrees), and the slopes of the lines formed by the points D, E, and F do not indicate a right angle.

However, we can also use the Pythagorean theorem to verify our conclusion. The Pythagorean theorem states that in a right 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.

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

  • Length of DE: √((-2 - (-2))^2 + (2 - (-1))^2) = √(0^2 + 3^2) = √9 = 3
  • Length of EF: √((0 - (-2))^2 + (0 - 2)^2) = √(2^2 + 2^2) = √8
  • Length of FD: √((-2 - 0)^2 + (-1 - 0)^2) = √((-2)^2 + (-1)^2) = √5

Now, let's calculate the square of the length of the hypotenuse (EF) and compare it to the sum of the squares of the lengths of the other two sides (DE and FD):

  • (EF)^2 = (√8)^2 = 8
  • (DE)^2 + (FD)^2 = 3^2 + (√5)^2 = 9 + 5 = 14

Since (EF)^2 β‰  (DE)^2 + (FD)^2, we can conclude that the triangle DEF is not a right triangle.

In conclusion, based on the analysis of the slopes of the lines formed by the points D, E, and F, and the Pythagorean theorem, we can conclude that the triangle DEF is not a right triangle.