In ΔABC, D and F are the midpoints of the sides AB and AC, respectively. E is a point on the segment DF such that DE:EF=1:2. If DE=4cm, then BC is equal to:
- A20 cm
- B26 cm
- C22 cm
- D24 cm
Solution & Step-by-step Explanation
According to the Midpoint Theorem in geometry, the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half of its length.
Since D and F are the midpoints of AB and AC respectively, we have:
DF=
2
1
BC⟹BC=2×DF
We are given that E lies on DF such that:
EF
DE
=
2
1
Given that DE=4cm, we can find EF:
EF
4
=
2
1
⟹EF=8cm
The total length of the segment DF is:
DF=DE+EF=4+8=12cm
Now, using the relation from the Midpoint Theorem:
BC=2×DF=2×12=24cm
Since D and F are the midpoints of AB and AC respectively, we have:
DF=
2
1
BC⟹BC=2×DF
We are given that E lies on DF such that:
EF
DE
=
2
1
Given that DE=4cm, we can find EF:
EF
4
=
2
1
⟹EF=8cm
The total length of the segment DF is:
DF=DE+EF=4+8=12cm
Now, using the relation from the Midpoint Theorem:
BC=2×DF=2×12=24cm