Final answer:
The correct option is A). After calculating the side lengths of triangles ABC and DEF using the coordinates provided, it is determined that both are isosceles right triangles with corresponding angles and proportional sides. Hence, triangle ABC is similar to triangle DEF.
Step-by-step explanation:
To determine if triangles ABC and DEF are similar, we need to compare their corresponding sides and angles. Similar triangles have all their corresponding angles equal and their corresponding sides are proportional. Using the coordinates given for triangle ABC (A (0,4), B (0,0), C (4,0)) and triangle DEF (D (1,-1), E (1,-2), F (2,-2)), we can calculate the lengths of their sides using the distance formula.
For triangle ABC, AB is a vertical segment with a length of 4 units, BC is a horizontal segment with a length of 4 units, and AC is the hypotenuse that can be calculated as the square root of (AB)^2 + (BC)^2, which gives us 4√2 units. These side lengths suggest that triangle ABC is an isosceles right triangle (45-45-90).
For triangle DEF, DE is a vertical segment with a length of 1 unit, EF is a horizontal segment with a length of 1 unit, and DF can be calculated in the same manner as AC, resulting in √2 units. Thus, triangle DEF is also an isosceles right triangle with each angle measuring 45 degrees and 90 degrees respectively.
Since the lengths of the sides of triangle DEF are proportional to the lengths of the sides of triangle ABC, and all corresponding angles are equal, we can conclude that triangle ABC is similar to triangle DEF. Therefore, the answer is A) Yes, triangle ABC is similar to triangle DEF.