Question

Which statement accurately describes triangle DEF with coordinates D(2, 1), E(3, 5), and F(6, 2)?

A. Triangle DEF is an equilateral triangle because all three sides have equal lengths.
B. Triangle DEF is not an equilateral triangle because it does not have congruent sides.
C. Triangle DEF is an equilateral triangle because it has two sides of equal length.
D. Triangle DEF is not an equilateral triangle because its angles are not all congruent.

Answer 1

Triangle DEF is not an equilateral triangle as it does not have all sides of equal length. The lengths DE and DF are both √17, but EF is √18, indicating that the sides are not congruent.

Step-by-step explanation:

To determine whether triangle DEF with coordinates D(2, 1), E(3, 5), and F(6, 2) is an equilateral triangle, we need to calculate the lengths of its sides using the distance formula. The distance formula is derived from the Pythagorean theorem and is used to find the distance between two points (x1, y1) and (x2, y2) in the coordinate plane: distance = √((x2 - x1)² + (y2 - y1)²).

Let's calculate each side of the triangle:

• Length DE = √((3 - 2)² + (5 - 1)²) = √(1² + 4²) = √17
• Length DF = √((6 - 2)² + (2 - 1)²) = √(4² + 1²) = √17
• Length EF = √((6 - 3)² + (2 - 5)²) = √(3² + (-3)²) = √18

Since the lengths of the sides DE and DF are equal, but the length EF is different, triangle DEF is not equilateral. An equilateral triangle must have all sides of equal length which is not the case here, thus we choose the option:

B. Triangle DEF is not an equilateral triangle because it does not have congruent sides.