Ray is a licensed technician in the Philippines. The loves to write about mathematics and also civil engineering.

You are watching: Converse of the triangle proportionality theorem

## What Is the Triangle Proportionality Theorem?

The triangle proportionality theorem says that if a heat parallel to one side of a triangle intersects the other two sides in different points, it divides the political parties into matching proportional segments.

Some geometry books call the triangle proportionality to organize the side-splitting theorem. The side-splitting theorem has actually the same summary as the triangle proportionality theorem. Lock coined together terms for the theorem due to the fact that of the midsegment that splits the intersecting side into two.

## Triangle Proportionality to organize Proof

How perform you prove this theorem? consider the triangle alphabet below. Permit D and E it is in points ~ above line abdominal and heat BC, respectively, such the line DE is parallel to heat AC. Let united state prove the proportion equation that BD/DA = BE/EC.

Triangle Proportionality organize Proof

StatementReasons

1. ∠BDE ≅ ∠BAC & ∠BED ≅ ∠BCA

1. Parallel lines kind corresponding congruent angles.

2. △ABC ~ △DBE

2. AA Similarity Theorem

3. BD/BA = BE/BC

3. Corresponding sides of similar triangles space proportional

4. BA/BD = BC/BE

4. Reciprocal Property

5. (BA – BD)/BD = (BC – BE)/BE

6. DA/BD = EC/BE or BD/DA = BE/EC

6. An easy Triangle Proportionality Theorem

John beam Cuevas

Triangle Proportionality Steps

How carry out you deal with proportional parts in triangles and also parallel lines?

1. Situate the parallel lines. Note that these 2 parallel lines intersect the two sides of the triangle and any next of the triangle.

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

3. Recognize the point out of consideration and also look because that the matching sides of comparable triangles.

Triangle Proportionality Formula

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

SS1/LS1 = SS2/LS2

Where:

SS1 = smaller △ next 1

LS1 = bigger △ side 1

SS2 = smaller sized △ side 2

LS2 = larger △ side 2

## The Converse the the Triangle Proportionality organize Proof

The converse of the triangle proportionality theorem says that if a heat intersects 2 sides of a triangle and also cuts off segments’ proportionality, that is parallel to the third. In △ABC, allow D and also E be points ~ above line ab and BC, respectively, such the BD/DA = BE/EC. Now, prove that line DE is parallel to line AC.

The Converse the the Triangle Proportionality theorem Proof

StatementReasons

1. AC // DE ∩ C’

1. Think about AC together a line with A parallel to line DE, intersecting next BC in ~ C’.

2. BD/DA = BE/EC’

2. Straightforward 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

The Converse the the Triangle Proportionality Theorem

John beam Cuevas

## Example 1: completing the Proportions

Given the following triangles, complete the proportions for the adjoining numbers using the triangle proportionality theorem. Take into consideration that in △PRQ, line ST is parallel to heat PQ.

a. RS/SP

b. TQ/RQ

Triangle Proportionality Theorem example 1: completing the Proportions

John beam Cuevas

Solution

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

RS/SP = RT/TQ

As you deserve to observe indigenous the provided in letter b, both TQ and RQ are components of the triangle PRQ. The line TQ corresponds with RQ. Thus, to complete the proportions the the provided adjoining lines, look because that the other equivalent intersected by the parallel heat in the triangle, i m sorry is heat SP and also line RP. Mental that corresponding sides of comparable triangles space proportional.

TQ/RQ = SP/RP

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

## Example 2: completing the Proportions because that Adjoining Figures

Complete the proportions because that the adjoining figures. Offered △ABC, consider line DE is parallel to line BC.

d. AE/AC

Completing the Proportions for Adjoining Figures

John beam Cuevas

Solution

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

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

## Example 3: finding the variable "X" using Triangle Proportionality Theorem

Find x in each of the figures below. Take keep in mind that numbers are not attracted to scale.

Finding the change "X" utilizing Triangle Proportionality Theorem

John ray Cuevas

Solution

By the straightforward triangle proportionality theorem, we have the complying with 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

The last answers space x = 15/2 and x = 8.

## Example 4: prove Proportion Formulas utilizing Triangle Proportionality Theorem

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

Proving ratio Formulas making use of Triangle Proportionality Theorem

John ray Cuevas

Solution

In the offered triangle ABC, heat DE is parallel to heat AC, and also D is the midpoint of line AB. It shows that DE intersects the other two sides at various points and also divides them into corresponding proportional segments. Come prove the E is the midpoint of BC, allow us present the perform of statements and reasons.

StatementReasons

1. DE // AC; D is the midpoint heat AB

1. Given

2. BD/DA = BE/EC

2. Simple Triangle Proportionality Theorem

3. BD = DA

3. Definition of a midpoint

4. BD/DA = BE/EC = 1

Substitution

5. It is in = EC

5. Cross - Product Property

6. E is the midpoint of heat BC.

6. An interpretation of a midpoint

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

## Example 5: using the Triangle Proportionality Theorem

In ∆ABC, heat DE is attracted parallel to line AC. Provided that abdominal = 12, DB = 4, and also BC = 24, discover CE.

Applying the Triangle Proportionality Theorem

John ray Cuevas

Solution

(AB - DB)/AB = CE/BC

(12 - 4)/12 = CE/24

CE = 16

The value of CE is 16 units.

## Example 6: producing a relationship Formula

If L1 is parallel come L2 and also x + y = 15, discover x and y.

Creating a proportion Formula

John ray Cuevas

Solution

Based ~ above the offered triangle, L1 intersects 2 sides that the triangle at 2 points, B and E. L1 is parallel to L2 in ~ points C and also D. Therefore, L1 divides the political parties of the triangle into equivalent proportional segments, thus, conforms the Triangle Proportionality Theorem. First, determine the values of the proportional parts of the triangle.

AC = 8 + 2

AC = 10

AB = 2

AE = x

Create a proportion formula because that the sides of the offered triangle and substitute the values derived earlier.

10/2 = 15/x

5 = 15/x

x = 3

Solve for the change y through substituting to the equation x + y = 15,

x + y = 15

3 + y = 15

y = 12

The worths of x and also y space 3 and also 12, respectively.

## Example 7: Applications the the Triangle Proportionality Theorem

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

Applications the the Triangle Proportionality Theorem

John beam Cuevas

Solution

As you have the right to observe, the building's height creates a appropriate triangle v its zero on the ground. To solve the x distance, the human being wants to stand in the shade from the building and also use the triangle proportionality theorem. Come start, transform 5 ft 6 inches come 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

If the human wants to was standing in the shade as far away indigenous the building as possible, he must stand 3.083 feet far from the building.

## Example 8: Triangle Proportionality Theorem native Problem

To find the height of a leg that connects 2 buildings, a male 6 feet high stands at one end and looks under to the ground at the various other end. Using the distances significant in the figure below, find the elevation of the bridge.

Triangle Proportionality Theorem indigenous Problem

John beam Cuevas

Solution

You can utilize triangle proportionality in this problem. Permit 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

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

## Example 9: utilizing the Triangle Proportionality Theorem

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

Utilizing the Triangle Proportionality Theorem

John beam Cuevas

Solution

The given triangle ABC has two various triangles that deserve to be used to analysis the Triangle Proportionality Theorem, the △ADC and △ABC. Use the given values AF = 4 and also FD = 6, and also create a proportionality equation.

AE/EC = AF/FD

AE/EC = 4/6

Create the proportionality formula for the more big triangle ABC. Because the value of AE/EC derived from the vault equation is 4/6, substitute this worth to the proportionality equation displayed below.

Finally, deal with the value of DB.

10/DB = 4/6

DB = <10(6)>/4

DB = 15

The worth of DB is 15 units.

## Example 10: recognize the absent Values making use of the Triangle Proportionality Theorem

Refer to the number below and also compute for the following. Assume that line DE is parallel line BC. A. If a = 5, abdominal muscle = 10, and also p = 12, uncover the value of q. B. If c = 5, AC = 15, and q = 24, discover the worth of p.C. If b = 9, ns = 21, q = 34, find the value of a.

Finding the absent Values utilizing the Triangle Proportionality Theorem

John ray Cuevas

Solution

Apply the Triangle Proportionality theorem on every question, as displayed 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

The final answers room q = 24, ns = 8, and a = 189/13.

This content is accurate and true come the ideal of the author’s knowledge and also is not intended to substitute because that formal and individualized advice indigenous a standard professional.

See more: How Much Is 25 Grams Of Butter To Tablespoons Conversion (G To Tbsp)

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