Intercept Theorem: Counter-Example & Proof

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In summary, the intercept theorem states that if three lines make equal intercepts on each of two transversals PQ and RS, then the three lines must be parallel. However, the converse to this theorem does not hold as there exists a counter-example. In order to produce a counter-example, a drawing with length measurements is necessary. However, the steps of proving this counter-example are not specified.
  • #1
lwymarie
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intercept theorem--urgent

does the converse to the intercept theorem hold? i.e. if three lines make equal intercepts on each of two transversals PQ and RS as shown, is it true that the three lines must be parallel? can you produce a counter-example? (a drawing indicated with length measurements in necessary)

I can draw the diagram of a counter-example but i don't know how to write the steps of proving it:(
 
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  • #2
lwymarie said:
does the converse to the intercept theorem hold? i.e. if three lines make equal intercepts on each of two transversals PQ and RS as shown, is it true that the three lines must be parallel? can you produce a counter-example? (a drawing indicated with length measurements in necessary)

I can draw the diagram of a counter-example but i don't know how to write the steps of proving it:(
Perhaps you could attach the drawing. The description of the problem is not clear.

AM
 
  • #3


Yes, the converse to the intercept theorem does hold. This can be proven using a counter-example.

Counter-example: Let's consider three lines, AB, CD, and EF, intersecting two transversals PQ and RS as shown in the diagram below.

<img src="" width="200">

We can see that AB, CD, and EF all make equal intercepts on both transversals PQ and RS. However, these three lines are not parallel.

Proof: To prove that the converse to the intercept theorem holds, we need to show that if three lines make equal intercepts on each of two transversals, then they must be parallel.

Let's label the points where the lines intersect the transversals as A, B, C, D, E, and F as shown in the diagram.

<img src="" width="200">

Since the intercept theorem states that the ratio of the intercepts is equal to the ratio of the intercepted segments, we can write the following equations:

AB/CD = PQ/RS
CD/EF = RS/PQ
EF/AB = PQ/RS

From these equations, we can see that AB/CD = EF/AB, which means that AB^2 = CD x EF.

Similarly, we can also see that CD/EF = AB/CD, which means that CD^2 = EF x AB.

Therefore, AB^2 = CD^2, which implies that AB = CD.

Similarly, we can also show that CD = EF and AB = EF.

This means that all three lines are equal in length, which is only possible if they are parallel.

Hence, we have proven that if three lines make equal intercepts on each of two transversals, they must be parallel, which is the converse to the intercept theorem.
 

Related to Intercept Theorem: Counter-Example & Proof

1. What is the Intercept Theorem?

The Intercept Theorem is a geometric principle that states that if two parallel lines are intersected by a transversal, then the segments intercepted on one line are proportional to the segments intercepted on the other line.

2. What is a counter-example for the Intercept Theorem?

A counter-example for the Intercept Theorem is a scenario where the theorem does not hold true. For example, if the parallel lines are not intersected by a transversal or if the segments intercepted on one line are not proportional to the segments intercepted on the other line.

3. How can the Intercept Theorem be proven?

The Intercept Theorem can be proven using several methods, including using the properties of similar triangles, using algebraic equations, or using the concept of ratio and proportion.

4. What are the applications of the Intercept Theorem?

The Intercept Theorem has various applications in geometry, such as in solving problems involving parallel and intersecting lines, similar triangles, and proportional segments. It is also useful in real-world scenarios, such as in construction and engineering.

5. How is the Intercept Theorem related to other geometric theorems?

The Intercept Theorem is related to other geometric theorems, such as the Angle Bisector Theorem and the Triangle Proportionality Theorem. These theorems all involve the concept of proportions and ratios in similar triangles.

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