Ray is a licensed technician in the Philippines. He loves come write around mathematics and also civil engineering.

You are watching: Converse of the triangle proportionality theorem

## What Is the Triangle Proportionality Theorem?

The triangle proportionality theorem claims that if a line parallel to one side of a triangle intersects the various other two political parties in various points, the divides the sides into corresponding proportional segments.

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

## Triangle Proportionality organize Proof

How carry out you prove this theorem? take into consideration the triangle abc below. Permit D and E be points top top line abdominal and line BC, respectively, such the line DE is parallel to line AC. Let united state prove the ratio equation the BD/DA = BE/EC.

Triangle Proportionality to organize Proof

StatementReasons

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 comparable triangles room proportional

4. BA/BD = BC/BE

4. Mutual Property

5. (BA – BD)/BD = (BC – BE)/BE

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

6. Simple Triangle Proportionality Theorem John beam Cuevas

Triangle Proportionality Steps

How perform you resolve proportional parts in triangles and parallel lines?

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

2. Identify the comparable triangles in the given number using the AA similarity theorem. The AA similarity theorem says that if 2 angles of one triangle are congruent to 2 angles of one more triangle, then the two triangles are similar.

3. Determine the points of consideration and look because that the equivalent sides of comparable triangles.

Triangle Proportionality Formula

There is no really formula for this theorem, however you can use variables/terms such together the following:

SS1/LS1 = SS2/LS2

Where:

SS1 = smaller sized △ next 1

LS1 = bigger △ side 1

SS2 = smaller sized △ next 2

LS2 = bigger △ next 2

## The Converse of the Triangle Proportionality to organize Proof

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

The Converse of the Triangle Proportionality theorem Proof

StatementReasons

1. AC // DE ∩ C’

1. Think about AC together a line through A parallel to line DE, intersecting side BC at C’.

2. BD/DA = BE/EC’

2. An easy Triangle Proportionality Theorem

3. BE/EC’ = BE/EC & EC’ = EC & C = C’

3. By hypothesis

4. DE // AC

4. The Converse the 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 because that 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 instance 1: perfect the Proportions

John ray Cuevas

Solution

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

RS/SP = RT/TQ

As you deserve to observe native the given in letter b, both TQ and also RQ are components of the triangle PRQ. The heat TQ synchronizes with RQ. Thus, to complete the proportions the the given adjoining lines, look because that the other equivalent intersected by the parallel heat in the triangle, i beg your pardon is heat SP and line RP. Psychic that equivalent sides of comparable triangles are proportional.

TQ/RQ = SP/RP

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

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

Complete the proportions for the adjoining figures. Offered △ABC, take into consideration line DE is parallel to heat BC.

d. AE/AC Completing the Proportions because that Adjoining Figures

John ray Cuevas

Solution

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

For letter b, due to the fact that corresponding political parties of similar triangles space proportional, AE/AC is equal to AD/AB.

## Example 3: finding the change "X" utilizing Triangle Proportionality Theorem

Find x in each of the figures below. Take note that numbers are not drawn to scale. Finding the change "X" utilizing Triangle Proportionality Theorem

John ray Cuevas

Solution

By the straightforward 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

The final answers room x = 15/2 and also x = 8.

## Example 4: prove Proportion Formulas making use of Triangle Proportionality Theorem

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

Proving proportion Formulas making use of Triangle Proportionality Theorem

John ray Cuevas

Solution

In the provided triangle ABC, heat DE is parallel to line AC, and D is the midpoint of line AB. It reflects that DE intersects the various other two political parties at various points and also divides lock into corresponding proportional segments. To prove the E is the midpoint that BC, allow us display the perform of statements and also 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. Be = EC

5. Overcome - Product Property

6. E is the midpoint of heat BC.

6. Definition of a midpoint

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

## Example 5: using the Triangle Proportionality Theorem

In ∆ABC, heat DE is drawn parallel to heat AC. Offered that abdominal muscle = 12, DB = 4, and BC = 24, find CE.

Applying the Triangle Proportionality Theorem

John beam 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 to L2 and x + y = 15, uncover x and y.

Creating a ratio Formula

John beam Cuevas

Solution

Based on the given triangle, L1 intersects 2 sides of the triangle at two points, B and E. L1 is parallel to L2 at points C and also D. Therefore, L1 divides the sides of the triangle into corresponding 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 for the political parties of the offered triangle and substitute the values acquired earlier.

10/2 = 15/x

5 = 15/x

x = 3

Solve because that the change y by substituting to the equation x + y = 15,

x + y = 15

3 + y = 15

y = 12

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

## Example 7: Applications of the Triangle Proportionality Theorem

A 24-ft high structure casts a 4-ft zero on level ground. A human being 5 ft 6 in tall desires to was standing in the shade as much away from the structure as possible. What is this distance?

Applications of the Triangle Proportionality Theorem

John beam Cuevas

Solution

As you can observe, the building's height forms a appropriate triangle v its zero on the ground. To solve the x distance, the person wants to was standing in the shade from the building and also use the triangle proportionality theorem. Come start, transform 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

If the human wants to stand in the the shade as far away from the structure as possible, he have to stand 3.083 feet away from the building.

## Example 8: Triangle Proportionality Theorem indigenous Problem

To uncover the elevation of a bridge that connects 2 buildings, a man 6 feet high stands in ~ one end and looks down to the ground at the various other end. Using the distances marked in the figure below, discover the height of the bridge.

Triangle Proportionality Theorem native Problem

John beam Cuevas

Solution

You can utilize triangle proportionality in this problem. Permit H it is in 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 leg connecting the two buildings is 15 feet.

## Example 9: making use of the Triangle Proportionality Theorem

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

Utilizing the Triangle Proportionality Theorem

John ray Cuevas

Solution

The offered triangle ABC includes two different triangles that deserve to be utilized to analysis the Triangle Proportionality Theorem, the △ADC and △ABC. Make use of the provided 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 because that the more big triangle ABC. Due to the fact that the worth of AE/EC derived from the vault equation is 4/6, substitute this value to the proportionality equation presented below.

We understand that ad is the sum of AF and also FD.

Finally, settle the value of DB.

10/DB = 4/6

DB = <10(6)>/4

DB = 15

The value of DB is 15 units.

## Example 10: finding the lacking Values using the Triangle Proportionality Theorem

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

Finding the absent Values using the Triangle Proportionality Theorem

John ray Cuevas

Solution

Apply the Triangle Proportionality to organize on every question, as presented 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 roughly 14.54

The last answers space q = 24, ns = 8, and also a = 189/13.

This content is accurate and also true come the best of the author’s knowledge and also is not expected to substitute for formal and individualized advice indigenous a default professional.

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