Final answer:
Points A, B, and C are midpoints of the sides of right triangle DEF. BC = 6 cm, AC = 5 cm, and BA = 4 cm are true statements, while the perimeter of triangle ABC = 12 cm and the area of triangle ABC is equal to the area of triangle DEF are false.
Step-by-step explanation:
The given information in the question states that points A, B, and C are midpoints of the sides of right triangle DEF. Let's analyze each statement:
BC = 6 cm:
Since point B is the midpoint of DE, it means that BC is half the length of DE. Therefore, if DE = 12 cm, BC = 6 cm.
AC = 5 cm:
Similarly, point C being the midpoint of EF implies that AC is half the length of EF. So, if EF = 10 cm, AC = 5 cm.
BA = 4 cm:
In this case, point A is the midpoint of DF, which means that BA is half the length of DF. If DF = 8 cm, BA = 4 cm.
The perimeter of triangle ABC = 12 cm:
The perimeter of a triangle is the sum of its side lengths. Since AB = 4 cm, BC = 6 cm, and AC = 5 cm, the perimeter of triangle ABC is 4 + 6 + 5 = 15 cm, not 12 cm. Therefore, this statement is false.
The area of triangle ABC is the area of triangle DEF:
The area of a triangle is given by the formula A = (1/2) * base * height. Since triangle ABC is smaller than triangle DEF, its base and height are also proportionally smaller. Therefore, the area of triangle ABC is not equal to the area of triangle DEF. Hence, this statement is false.
Based on the analysis above, the true statements are BC = 6 cm, AC = 5 cm, and BA = 4 cm.