# Triangle Proportionality Theorem (With Proof and Examples)

*Ray is a licensed engineer in the Philippines. He loves to write about mathematics and civil engineering.*

## What Is the Triangle Proportionality Theorem?

The triangle proportionality theorem states that if a line parallel to one side of a triangle intersects the other two sides in different points, it divides the sides into corresponding proportional segments.

Some geometry books call the triangle proportionality theorem the side-splitting theorem. The side-splitting theorem has the same description as the triangle proportionality theorem. They coined such terms for the theorem because of the midsegment that splits the intersecting side into two.

## Triangle Proportionality Theorem Proof

How do you prove this theorem? Consider the triangle ABC below. Let D and E be points on line AB and line BC, respectively, such that line DE is parallel to line AC. Let us prove the proportion equation of BD/DA = BE/EC.

Statement | Reasons |
---|---|

1. ∠BDE ≅ ∠BAC & ∠BED ≅ ∠BCA | 1. Parallel lines form corresponding congruent angles. |

2. △ABC ~ △DBE | 2. AA Similarity Theorem |

3. BD/BA = BE/BC | 3. Corresponding sides of similar triangles are proportional |

4. BA/BD = BC/BE | 4. Reciprocal Property |

5. (BA – BD)/BD = (BC – BE)/BE | 5. Denominator Addition/Subtraction Property |

6. DA/BD = EC/BE or BD/DA = BE/EC | 6. Basic Triangle Proportionality Theorem |

**Triangle Proportionality Steps**

How do you solve proportional parts in triangles and parallel lines?

1. Locate the parallel lines. Note that these two parallel lines intersect the two sides of the triangle and any side of the triangle.

2. Identify the similar triangles in the given figure using the AA similarity theorem. The AA similarity theorem states that if two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.

3. Identify the points of consideration and look for the corresponding sides of similar triangles.

**Triangle Proportionality Formula**

There is no actual formula for this theorem, but you can use variables/terms such as the following:

SS1/LS1 = SS2/LS2

Where:

SS1 = Smaller △ Side 1

LS1 = Larger △ Side 1

SS2 = Smaller △ Side 2

LS2 = Larger △ Side 2

## The Converse of the Triangle Proportionality Theorem Proof

The converse of the triangle proportionality theorem states that if a line intersects two sides of a triangle and cuts off segments’ proportionality, it is parallel to the third. In △ABC, let D and E be points on line AB and BC, respectively, such that BD/DA = BE/EC. Now, prove that line DE is parallel to line AC.

Statement | Reasons |
---|---|

1. AC // DE ∩ C’ | 1. Consider AC as a line through A parallel to line DE, intersecting side BC at C’. |

2. BD/DA = BE/EC’ | 2. Basic Triangle Proportionality Theorem |

3. BE/EC’ = BE/EC & EC’ = EC & C = C’ | 3. By hypothesis |

4. DE // AC | 4. The Converse of the Triangle Proportionality Theorem |

## Example 1: Completing the Proportions

Given the following triangles, complete the proportions for the adjoining figures using the triangle proportionality theorem. Consider that in △PRQ, line ST is parallel to line PQ.

a. RS/SP

b. TQ/RQ

**Solution**

For letter a, given that ST is a line parallel to the side PQ, and it intersects the other two sides RP and RQ into two different points, we can conclude that line ST divides the sides into corresponding proportional segments. With the triangle proportionality theorem, then RS/SP is equal to RT/TQ.

RS/SP = RT/TQ

As you can observe from the given in letter b, both TQ and RQ are parts of the triangle PRQ. The line TQ coincides with RQ. Thus, to complete the proportions of the given adjoining lines, look for the other counterpart intersected by the parallel line in the triangle, which is line SP and line RP. Remember that corresponding sides of similar triangles are proportional.

TQ/RQ = SP/RP

**Answer**

RS/SP = RT/TQ and TQ/RQ = SP/RP

## Example 2: Completing the Proportions for Adjoining Figures

Complete the proportions for the adjoining figures. Given △ABC, consider line DE is parallel to line BC.

c. DB/AD

d. AE/AC

**Solution**

Given that the line DE is parallel to line BC, triangle proportionality theorem line DB/AD is proportional to EC/AE.

DB/AD = EC/AE

For letter b, since corresponding sides of similar triangles are proportional, AE/AC is equal to AD/AB.

AE/AC = AD/AB

**Answer**

DB/AD = EC/AE and AE/AC = AD/AB

## Example 3: Finding the Variable "X" Using Triangle Proportionality Theorem

Find x in each of the figures below. Take note that figures are not drawn to scale.

**Solution**

By the basic triangle proportionality theorem, we have the following solutions:

6/4 = (x - 3)/3

4(x - 3) = 18

4x - 12 = 18

4x = 18 + 12

4x = 30

x = 15/2

x/4 = 16/x

x^2 = 64

x = 8

**Answer**

The final answers are x = 15/2 and x = 8.

## Example 4: Proving Proportion Formulas Using Triangle Proportionality Theorem

In ∆ABC, line DE is parallel to line AC. D is the midpoint of line AB. Prove that E is the midpoint of BC.

**Solution**

In the given triangle ABC, line DE is parallel to line AC, and D is the midpoint of line AB. It shows that DE intersects the other two sides at different points and divides them into corresponding proportional segments. To prove that E is the midpoint of BC, let us show the list of statements and reasons.

Statement | Reasons |
---|---|

1. DE // AC; D is the midpoint line AB | 1. Given |

2. BD/DA = BE/EC | 2. Basic Triangle Proportionality Theorem |

3. BD = DA | 3. Definition of a midpoint |

4. BD/DA = BE/EC = 1 | Substitution |

5. BE = EC | 5. Cross - Product Property |

6. E is the midpoint of line BC. | 6. Definition of a midpoint |

**Answer**

In ∆ABC, line DE is parallel to line AC. D is the midpoint of line AB and E is the midpoint of line BC.

## Example 5: Applying the Triangle Proportionality Theorem

In ∆ABC, line DE is drawn parallel to line AC. Given that AB = 12, DB = 4, and BC = 24, find CE.

**Solution**

AD/AB = CE/BC

(AB - DB)/AB = CE/BC

(12 - 4)/12 = CE/24

CE = 16

**Answer**

The value of CE is 16 units.

## Example 6: Creating a Proportion Formula

If L_{1} is parallel to L_{2} and x + y = 15, find x and y.

**Solution**

Based on the given triangle, L_{1} intersects two sides of the triangle at two points, B and E. L_{1} is parallel to L_{2} at points C and D. Therefore, L_{1} divides the sides of the triangle into corresponding proportional segments, thus, conforms the Triangle Proportionality Theorem. First, identify the values of the proportional parts of the triangle.

AC = 8 + 2

AC = 10

AB = 2

AD = x + y

AD = 15

AE = x

Create a proportion formula for the sides of the given triangle and substitute the values obtained earlier.

AC/AB = AD/AE

10/2 = 15/x

5 = 15/x

x = 3

Solve for the variable y by substituting to the equation x + y = 15,

x + y = 15

3 + y = 15

y = 12

**Answer**

The values of x and y are 3 and 12, respectively.

## Example 7: Applications of the Triangle Proportionality Theorem

A 24-ft high building casts a 4-ft shadow on level ground. A person 5 ft 6 in tall wants to stand in the shade as far away from the building as possible. What is this distance?

**Solution**

As you can observe, the building's height forms a right triangle with its shadow on the ground. To solve the x distance, the person wants to stand in the shade from the building and use the triangle proportionality theorem. To start, convert 5 ft 6 inches to feet.

5’6” = 5.5 feet

24/4 = 5.5/4-x

6 = 5.5/4-x

6 (4 - x) = 5.5

24 - 6x = 5.5

24 - 5.5 = 6x

6x = 18.5

x = 3.083 feet

**Answer**

If the person wants to stand in the shade as far away from the building as possible, he must stand 3.083 feet away from the building.

## Example 8: Triangle Proportionality Theorem Word Problem

To find the height of a bridge that connects two buildings, a man 6 feet tall stands at one end and looks down to the ground at the other end. Using the distances marked in the figure below, find the height of the bridge.

**Solution**

You can utilize triangle proportionality in this problem. Let H be the height of the bridge.

(H + 6) / 35 = H / 25

25 (H + 6) = 35H

25H +150 = 35H

35H - 25H = 150

10H = 150

H = 15 feet

**Answer**

The height of the bridge connecting the two buildings is 15 feet.

## Example 9: Utilizing the Triangle Proportionality Theorem

In ∆ABC, DE // BC, FE // DC, AF = 4, and FD = 6. Find DB.

**Solution**

The given triangle ABC contains two different triangles that can be utilized to analyze the Triangle Proportionality Theorem, the △ADC and △ABC. Utilize the given values AF = 4 and FD = 6, and create a proportionality equation.

AE/EC = AF/FD

AE/EC = 4/6

Create the proportionality formula for the more big triangle ABC. Since the value of AE/EC obtained from the previous equation is 4/6, substitute this value to the proportionality equation shown below.

AD/DB = AE/EC

AD/DB = 4/6

We know that AD is the sum of AF and FD.

AD = AF + FD

AD = 4 + 6

AD = 10

Finally, solve the value of DB.

10/DB = 4/6

DB = [10(6)]/4

DB = 15

**Answer**

The value of DB is 15 units.

## Example 10: Finding the Missing Values Using the Triangle Proportionality Theorem

Refer to the figure below and compute for the following. Assume that line DE is parallel line BC.

a. If a = 5, AB = 10, and p = 12, find the value of q.

b. If c = 5, AC = 15, and q = 24, find the value of p.

C. If b = 9, p = 21, q = 34, find the value of a.

**Solution**

Apply the Triangle Proportionality Theorem on each question, as shown below.

a/p = AB/q

5/12 = 10/q

q = 24

c/p = AC/q

5/p = 15/24

p = 8

a/p = (a + b)/q

a/21 = (a + 9)/34

a = 189/13 or approximately 14.54

**Answer**

The final answers are q = 24, p = 8, and a = 189/13.

*This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.*

**© 2021 Ray**