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Points A, B, and C are midpoints of the sides of right triangle DEF.

Triangle A B C is inside triangle D E F. Point A is the midpoint of side F D, point B is the midpoint of side D E, point C is the midpoint of side F E. Angles D F E and A B C are right angles. The length of D E is 10 centimeters, the length of F D is 6 centimeters, and the length of F E is 8 centimeters.


Which statements are true? Select three options. (The formula for the area of a triangle is A = One-halfbh.)

BC = 6 cm
AC = 5 cm
BA = 4 cm
The perimeter of triangle ABC = 12 cm.
The area of triangle ABC is One-third the area of triangle DEF.

All Answers 1

Answered by GPT-5 mini AI
True: AC = 5 cm; BA = 4 cm; the perimeter of triangle ABC = 12 cm.

Reason: A, B, C are midpoints, so each side of ABC is a midsegment = half the opposite side of DEF.
- BC = 1/2 · FD = 1/2 · 6 = 3 (so BC = 6 is false).
- AC = 1/2 · DE = 1/2 · 10 = 5 (true).
- BA = 1/2 · FE = 1/2 · 8 = 4 (true).
Perimeter = 3 + 5 + 4 = 12 (true).

Area: area(DEF) = (1/2)(6)(8) = 24, area(ABC) = (1/2)^2 · 24 = 6, so area(ABC) = 1/4 area(DEF), not 1/3 (false).