One parallel cut, two proportional divisions
A line parallel to one side of a triangle divides the other two sides in the same ratio. The parallel condition is essential.
Learning goals
- State BPT including its parallel-line condition.
- Use BPT to form and solve correct segment proportions.
- Use converse BPT to establish parallel lines.
- Reproduce the BPT proof using justified area ratios.
Concept lesson
Parallel lines create equal angles
If DE is parallel to BC in triangle ABC, corresponding angles ADE and ABC are equal, and AED and ACB are equal.
Solve a proportion
Keep like-position segments together, then cross multiply to solve an unknown ratio term.
BPT statement and diagram conditions
In triangle ABC, if D lies on AB, E lies on AC and DE is parallel to BC, then AD/DB = AE/EC. Confirm all three conditions before invoking BPT.
Converse BPT reverses the implication
If D and E divide AB and AC in the same ratio, AD/DB = AE/EC, then DE is parallel to BC. Use the converse only when proportional division is given or proved.
Proof architecture
A four-mark theorem response should name the given and to-prove statements, add the required perpendicular construction, establish two area ratios with reasons, use equal-base/equal-altitude area facts, and finish with the required proportion.
Form the BPT proportion before calculating
With DE parallel to BC, AD=3, DB=2 and AE=4.5, use AD/DB=AE/EC: 3/2=4.5/EC, hence EC=3. Label the small and remaining segments before substituting.
Prove a segment parallel by converse BPT
If AD=4, DB=6, AE=2 and EC=3, then AD/DB=4/6=2/3 and AE/EC=2/3. Since the two sides are divided proportionally, DE is parallel to BC by converse BPT.
BPT proof through areas
Construct perpendiculars from E to AB and from D to AC. Express area ratios ar(ADE)/ar(BDE)=AD/DB and ar(ADE)/ar(CDE)=AE/EC. Since triangles BDE and CDE lie on the same base DE and between the same parallels DE and BC, their areas are equal. Equating the ratios gives AD/DB=AE/EC.
Can BPT be used?
For four diagrams, verify point-on-side and parallel conditions; select only those where BPT is valid and state the resulting proportion.
Segment or whole side?
Correct deliberately mismatched proportions, convert whole-side data such as AB into DB=AB-AD where needed, then calculate the unknown.
Choose BPT or converse BPT
Classify prompts by direction: parallel given means use BPT to obtain ratios; equal ratios given means use converse BPT to obtain parallelism. Complete one proof of each type.
Parallel brace in a triangular frame
Use a brace known parallel to the base to calculate an unknown division; then use the resulting equal ratios to verify how the converse would certify parallel placement in a new frame.
Four-mark BPT proof checklist
State the theorem precisely; show the construction; write both area ratios with common-altitude reasons; justify equal areas between parallels; equate and conclude. Do not omit the reason beside each equality.
BPT mastery check
State BPT, solve one segment problem, prove one pair of lines parallel using the converse, and complete a reason-tagged area proof of BPT.