Question

In triangle ABC, segment DE is parallel to segment BC and segment EF is parallel to AB. Point D is on side AB, point E is on side AC, and point F is on side BC. Segment AD is 12, segment AE is 15, segment EC is 25, and segment FC is 30. According to the given theorem, which statement can be proved true?

1) Segment BF = 9
2) Segment BD = 18
3) Segment BD = 20
4) Segment BF = 24

Answer 1

To determine the length of segment BF in triangle ABC, the length of segment BD needs to be found first. Using the given lengths of segments AD, AE, EC, and FC, we can solve for BD using the concept of proportions. Once BD is known, it can be used to find the length of segment BF. The correct statement is that Segment BF = 24.

Step-by-step explanation:

According to the given information, we are given that segment DE is parallel to segment BC and segment EF is parallel to AB. We are also given the lengths of segments AD, AE, EC, and FC. To determine the length of segment BF (as mentioned in option 1), we need to find the length of segment BD first. Since triangle ABC is a similar triangle to triangle DBC, we can use the concept of proportions to find BD and then use it to find BF. Let's solve the problem step-by-step:

1. Use the proportions: AD/BD = AE/EC = DE/FC to find the length of segment BD. Substitute the given values into the equation: 12/BD = 15/25 = DE/30. Solve for BD to find: BD = (12 * 25)/15 = 20.
2. Now that we know BD = 20, we can use another proportion: BF/FC = BD/EC to find the length of segment BF. Substitute the known values into the equation: BF/30 = 20/25. Solve for BF to find: BF = (20 * 30)/25 = 24.
3. Therefore, the statement Segment BF = 24 is true.