Complete question:
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.
Answer:
2. A-C = 5 cm
3. B-A = 4 cm
4. The perimeter of triangle A-B-C = 12 cm.
Explanation:
First, picture a triangle formed by points D-E-F.
- Side F-D = 6 centimeters
- Side D-E = 10 centimeters
- Side F-E = 8 centimeters
Now,
- the midpoint of side F-D is A ⇒ 6/2 = 3cm
- the midpoint of side D-E is B ⇒ 10/2 = 5cm
- the midpoint of side F-E is C ⇒ 8/2 = 4cm
These values suggest that
- Side F-A ⇒ 3 cm = Side B-C ⇒ 3cm
- Side E-B ⇒ 5 cm = Side A-C ⇒ 5cm
- Side F-C = E-C ⇒ 4cm = Side A-B ⇒ 4cm
The perimeter of the interior triangle ABC = 3cm + 5cm + 4cm = 12 cm
The total area of the interior triangle ABC is 1/4 the area of triangle DEF. You can see in the picture that there are four equal triangles inside DEF.