Question

Which statement correctly describes whether Δdef is a right triangle?

1) Δdef is a right triangle because de¯¯¯¯¯ is perpendicular to df¯¯¯¯¯.
2) Triangle def is a right triangle because segment de is perpendicular to segment df.
3) Δdef is a right triangle because de¯¯¯¯¯ is perpendicular to ef¯¯¯¯¯.
4) Triangle def is a right triangle because segment de is perpendicular to segment ef.
5) Δdef is not a right triangle because no two sides are perpendicular.
6) Triangle def is not a right triangle because no two sides are perpendicular.
7) Δdef is a right triangle because df¯¯¯¯¯ is perpendicular to ef¯¯¯¯¯.

Answer 1

To ascertain whether ∆def is a right triangle, we must find a 90-degree angle between any two of its sides or validate if the Pythagorean theorem holds true for its sides. If there is no perpendicularity between any sides or the Pythagorean theorem conditions are not met, it cannot be a right triangle.

Step-by-step explanation:

To determine whether ∆def is a right triangle, one must identify if any two sides of the triangle are perpendicular to each other. A right triangle is characterized by a 90-degree angle between two of its sides. When assessing the statements given, we look for the one that describes a perpendicular relationship between the sides of ∆def:

1. ∆def is a right triangle because de is perpendicular to df.
2. ∆def is a right triangle because de is perpendicular to ef.
3. ∆def is a right triangle because df is perpendicular to ef.

We can also use the Pythagorean theorem, which applies specifically to right triangles. The theorem states that in a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse (a² + b² = c²). Therefore, if the squares of the lengths of two sides of ∆def equal the square of the length of the third side, we have a right triangle.

If none of these conditions are met, then ∆def cannot be a right triangle, which aligns with the statements that claim no two sides of the triangle are perpendicular.