Proportionality Theorems

What's a birthday party without cake! Most people are always excited to get to the end of the party because they get to eat cake. But not everybody knows that to bake a sweet cake, you need the right proportions of different ingredients. Your recipe may require 2 pounds of flour for one cake. That means if you want to bake two cakes, you'll need 4 pounds of flour. In maths, you can express this in form of a ratio and say the ratio of flour to one cake is 21 and if you were to bake two cakes, it will be 42 . These ratios are proportional to each other because they are equal.

Proportionality Theorems Proportionality Theorems

Create learning materials about Proportionality Theorems with our free learning app!

  • Instand access to millions of learning materials
  • Flashcards, notes, mock-exams and more
  • Everything you need to ace your exams
Create a free account
Contents
Table of contents

    What are Proportionality Theorems in Geometry?

    Proportionality theorems show relationships between shapes in the form of ratios. They show how different ratios of a figure or a quantity are equal. The proportionality theorems are mostly used in triangles. Let's look at the fundamental concept of the proportionality theorem using the triangle figures below.

    Proportionality Theorems Similar triangles StudySmarterSimilar Triangles - StudySmarter Originals

    The triangles above will be called similar triangles if their angles are congruent and if their corresponding sides are proportional. So, the proportionality formula for similar triangles is below.

    ABKL = ACKM = BCLM

    What is the Basic Proportionality Theorem?

    The Basic Proportionality Theorem focuses on showing the relationship between the length of the sides of a triangle.

    The proportionality theorem states that if a line is drawn parallel to one side of a triangle to intersect the other two sides at distinct points, then the other two sides are divided in the same ratio.

    The figure below gives a visual representation of the theorem.

    Proportionality Theorems Triangle StudySmarterA Triangle - StudySmarter Originals

    In the ABC above, DE¯ is parallel to BC¯. According to the Basic Proportionality Theorem, the ratio of AD¯ to DB is equal to the ratio of AE¯ to EC:

    ADDB = AEEC

    The ratio above is considered the basic proportionality formula.

    We can prove this theorem and find out how to get the formula. Let's see how.

    From the theorem, we know that DB and EC are in the same ratio and we want to prove that they are equal. We will first form triangles that have DB and EC as their side lengths. To get these triangles, we will draw a segment joining B to E and another segment joining C to D as shown below.

    Proportionality Theorems Triangle divided by segments StudySmarterA triangle divided into parts with segments - StudySmarter Originals

    We have now formed two new triangles(DEB and DEC).

    The next thing is to find a relationship between the new triangles. In particular, let's look at the area. DEB and DEC have the same base DE¯ and the same height because the third vertex of the triangle is between the same parallel. Therefore, the area of both triangles must be equal:

    Area(DEB) = Area(EDC)

    Now, consider AED. Let's take AD as the base and the height as the perpendicular distance from the line AD to the opposite vertex E. See how it looks like in the figure below.

    Proportionality Theorems Triangle divided by segments StudySmarterA triangle divided into parts with segments - StudySmarter Originals

    The area of this triangle is

    Area(AED) = 12 ×AD¯×EP¯

    We also need the area of DEB which will be:

    Area(DEB) = 12×DB¯×EP¯

    Now, we can take the ratio of the area of DEB to the area of AED and compare it with the ratio of the area of ECD to the area of AED. Therefore, the ratio of the areas is:

    ar(AED)ar(DEB) = 12×AD×EP12×DB×EP = ADDB

    As you can see, we've got the first part of the formula. To get the other, we will repeat everything we just did but withEDC.

    Unlike before, instead of using AD as the base of AED, we will use AE as the base and the height will be the perpendicular distance opposite the vertex D. See how it looks like in the figure below.

    Proportionality Theorems Triangle divided by segments StudySmarterA triangle divided into parts with segments - StudySmarter Originals

    The area of AED according to the image above is

    Area(AED) = 12×AE¯×DQ¯

    Now let's consider the area of EDC. We will take EC as the base and DQ as the height. The area is as follows.

    Area(EDC) = 12×EC¯×DQ¯

    We will now get the ratio of both areas to be:

    Area(AED)Area(EDC) = 12×AE×DQ12×EC×DQ = AEEC

    So, you can see that we've gotten the other part of the formula. But how do we show that both parts are equal? Let's equate both ratios and see.

    Area(AED)Area(DEB) = Area(AED)Area(EDC)

    Both numerators are the same so they are equal. Recall that at the beginning of the proof, we saw that

    Area(DEB) = Area(EDC)

    Therefore,

    ADDB = AEEC

    The Triangle Proportionality Theorem and Fundamental Theorem of Proportionality

    The Triangle Proportionality Theorem of Fundamental Theorem of Proportionality are just other names for the Basic Proportionality Theorem. You may see this theorem referred to as any of these titles!

    Proportionality Theorem Examples

    Let's see the application of the proportionality theorem with some examples.

    Consider a ABC where DE is parallel to BC. AD = 1.5cm, DB = 3cm, AE = 1. Find EC.

    Remember the formula

    ADDB = AEEC

    All we have to do is substitute the values.

    1.53 = 1EC1.5×EC = 3×11.5EC = 3EC = 31.5EC = 2cm

    Let's take a look at another example.

    Consider EFG where HL and EF are parallel to each other. EH = 9cm, HG = 21, FL = 6cm. Find LG

    Proportionality Theorems An example for solving proportionality StudySmarterrtio

    According to the proportionality theorem,

    EHHG = FLLG

    Subbing in the known values leaves us with

    921 = 6LG9×LG = 6×219LG = 126LG = 1269LG = 14 cm

    Aside from showing the relationship between the length of sides of triangles, in real life, the proportionality theorem can be used in construction.

    The converse of the Basic Proportionality Theorem

    The converse of the basic proportionality theorem is the reverse of the basic proportionality theorem. The theorem states that if a line is drawn to intersect two sides of a triangle at different points such that it cuts the two sides in the same ratio, then the line is parallel to the third side.

    Proportionality Theorems Triangle proportionality theorem StudySmarterA Triangle - StudySmarter Originals

    In the basic proportionality theorem, we saw that DE and BC are parallel and now we want to prove that DE and BC are indeed parallel. We would do this using the basic proportionality theorem which is

    ADDB = AEEC

    This proof is proof by contradiction meaning that we will assume that our desired result is wrong. We will assume that DE is not parallel to BC ((DEBC). If this is the case, then there must be another point on line AC such that a segment drawn from point D to that point is parallel to BC. See the figure below for clarity.

    Proportionality theorems Converse proportionality theorem StudySmarterA triangle divided into parts with segments - StudySmarter Originals

    Now that we have a line segment AF that is parallel to BC, we can now use the basic proportionality theorem which is

    ADDB = AFFC

    If you consider the basic proportionality theorem, you will have:

    ADDB = AFFC = AEEC

    We have now derived that DF is parallel to BC and we want to show that DE is parallel to BC. This means that what we really want to do is show that DF and DE are the same segments. So, if we consider the above equation, you will see that the first ratio is not really needed. So we are left with

    AFFC = AEEC

    We are now saying that DF and DE are the same segments which means that point F and point E are the same. If this is our conclusion, then the segment AF and AE are the same but we haven't really proven that yet.

    From the figure, we can say that the segment AC is equal to the sum of the segment AE and EC.

    AC = AE + EC

    Let's go back to one of our equations.

    AFFC = AEEC

    We will now add 1 (one) to both sides of the equation and bring them into the fractions by giving them a common denominator.

    AFFC + 1 = AEEC + 1AFFC + FCFC = AEEC + ECECAF + FCFC = AE + ECEC

    Both numerators on both sides of the equation are representations of the segment AC. So, we can replace them with AC

    ACFC = ACEC

    Let's simplify further by multiplying both sides by 1AC.

    ACFC × 1AC =ACEC ×1AC1FC =1EC

    Since they are equal, their reciprocals will also be equal. Therefore,

    FC¯ = EC¯

    You should observe that FC and EC are on the same line. If they are on the same line, the only way they can be equal is if both segments start at the same point. This means that point F must be equal to point E. It also means that the segment DE is the same as DF.

    This concludes that DF is indeed parallel to BC.

    Proportionality Theorems - Key takeaways

    • The basic proportionality theorem states that if a line is drawn parallel to one side of a triangle to intersect the other two sides at distinct points, then the other two sides are divided in the same ratio. The figure below gives a visual representation of the theorem.
    • The basic proportionality theorem is also referred to as the triangle proportionality theorem and proportionality segment theorem.
    • The converse of the basic proportionality theorem is the reverse of the basic proportionality theorem. The theorem states that if a line is drawn to intersect two sides of a triangle at different points such that it cuts the two sides in the same ratio, then the line is parallel to the third side.
    Frequently Asked Questions about Proportionality Theorems

    What is proportionality theorem?

    Proportionality theorems are for the purpose of showing relationships in form of ratios. They show how different ratios of a figure or a quantity are equal. 

    How do you solve proportionality theorem?

    You don't solve the theorem, you use the theorem to solve problems.

    What is the basic proportionality theorem formula?

    The basic proportionality theorem formula when there is a triangle ABC with a parallel segment DE is AD/DB = AE/EC

    What is the proportionality segment theorem?

    The proportionality segment theorem is the same as the basic proportionality theorem.

    How can the proportionality theorem be useful to us?

    The proportionality theorem is used to show the relationship between the length of sides of triangles. In real life, it can be used in construction.

    Test your knowledge with multiple choice flashcards

    The basic proportionality theorem is also called the triangle proportionality theorem.

    The converse proportionality theorem is the reverse of the basic proportionality theorem.

    The proportionality theorem can be used in construction. True or False

    Next

    Discover learning materials with the free StudySmarter app

    Sign up for free
    1
    About StudySmarter

    StudySmarter is a globally recognized educational technology company, offering a holistic learning platform designed for students of all ages and educational levels. Our platform provides learning support for a wide range of subjects, including STEM, Social Sciences, and Languages and also helps students to successfully master various tests and exams worldwide, such as GCSE, A Level, SAT, ACT, Abitur, and more. We offer an extensive library of learning materials, including interactive flashcards, comprehensive textbook solutions, and detailed explanations. The cutting-edge technology and tools we provide help students create their own learning materials. StudySmarter’s content is not only expert-verified but also regularly updated to ensure accuracy and relevance.

    Learn more
    StudySmarter Editorial Team

    Team Math Teachers

    • 9 minutes reading time
    • Checked by StudySmarter Editorial Team
    Save Explanation

    Study anywhere. Anytime.Across all devices.

    Sign-up for free

    Sign up to highlight and take notes. It’s 100% free.

    Join over 22 million students in learning with our StudySmarter App

    The first learning app that truly has everything you need to ace your exams in one place

    • Flashcards & Quizzes
    • AI Study Assistant
    • Study Planner
    • Mock-Exams
    • Smart Note-Taking
    Join over 22 million students in learning with our StudySmarter App

    Get unlimited access with a free StudySmarter account.

    • Instant access to millions of learning materials.
    • Flashcards, notes, mock-exams, AI tools and more.
    • Everything you need to ace your exams.
    Second Popup Banner