Proof: Parallel lines divide triangle sides proportionally | Similarity | Geometry | Khan Academy

Summary

This video script explores the geometric concept that if a line is parallel to one side of a triangle, it divides the other two sides proportionally. The proof involves establishing the similarity between triangles AED and ACB by using parallel lines and corresponding angles, ultimately leading to the proportional relationship between segment lengths AE/EC and AD/DB.

Takeaways

• 📐 The problem is to prove that if a line is parallel to one side of a triangle, it divides the other two sides proportionally.
• 🔍 A diagram is provided to help visualize the problem, with segment ED parallel to segment CB.
• 🔄 The concept of proportional division is introduced, stating that the ratio of the lengths of segments AE to EC should equal the ratio of AD to DB.
• 📚 To approach the proof, the instructor suggests establishing similarity between triangle AED and triangle ACB.
• 🔍 The parallel lines and transversal AC are used to identify congruent angles, which are key to proving similarity.
• 🔄 Angles one and three are congruent, as are angles two and four, due to being corresponding angles formed by the transversal.
• 🔄 Triangle AED is similar to triangle ACB by angle similarity, which is a crucial step in the proof.
• 🔄 The similarity of the triangles allows setting up a proportion between the lengths of segments AE and AC, and AD and AB.
• 📚 Algebraic manipulation is used to rearrange the proportion into a form that directly compares the lengths AE, EC, AD, and DB.
• 🔄 By cross-multiplying and simplifying, the instructor derives the equation AE * DB = AD * EC, which shows the proportional division.
• 🔄 Finally, dividing both sides of the equation by EC and DB respectively, confirms the proportionality of AE to EC and AD to DB, completing the proof.

Q & A

What is the main objective of the video script?

The main objective of the video script is to prove that if a line is parallel to one side of a triangle, it divides the other two sides proportionally.

What is the initial condition given in the script?

The initial condition given in the script is that segment ED is parallel to segment CB.

What does it mean for a line to divide the other two sides of a triangle proportionately?

For a line to divide the other two sides of a triangle proportionately means that the ratio of the lengths of the segments on one side of the dividing line to the lengths on the other side is the same for both sides being intersected.

How is the proportionality of the division expressed mathematically in the script?

The proportionality is expressed mathematically as the length of segment AE over the length of segment EC being equal to the length of segment AD over the length of segment DB.

What geometric property is used to establish similarity between triangle AED and triangle ACB?

The geometric property used to establish similarity between triangle AED and triangle ACB is the congruence of corresponding angles, which is a result of the parallel lines intersected by a transversal.

Why are the angles in triangles AED and ACB congruent?

The angles in triangles AED and ACB are congruent because they are corresponding angles formed by a transversal intersecting two parallel lines.

How does the script use the concept of similarity to prove the proportional division?

The script uses the concept of similarity by showing that triangles AED and ACB are similar due to congruent corresponding angles, which allows setting up a proportion between the lengths of the sides of these triangles.

What algebraic manipulation is performed to simplify the proportion set up in the script?

The algebraic manipulation performed includes cross-multiplying and then subtracting AE times AD from both sides of the equation to isolate the terms AE times DB and AD times EC.

What is the final equation derived from the algebraic manipulation in the script?

The final equation derived is AE times DB equals AD times EC, which shows the proportionality between the segments AE and EC, and AD and DB.

How does dividing both sides of the final equation by EC and DB prove the proportional division of the sides?

Dividing both sides by EC and DB cancels out these terms, leaving the ratio of AE to EC equal to the ratio of AD to DB, which is the desired proof of proportional division.