Proportionality Theorems for Triangles - Geometry Insights (2024)

In this lesson, the similarity of triangles will be used to prove some claims about triangles.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Try your knowledge on these topics.

Which of the following conditions guarantee that two triangles are similar?

Challenge

Solving Problems Using Triangle Similarity

The street lamps are 15 and 10 feet tall. How tall is the Grim Reaper?

Explore

Investigating Triangle Proportionality

Move the point on the side of the triangle. The applet draws a line parallel to another side of the triangle and gives the length of four line segments.

  • Move the vertices and investigate the relationship between these lengths. What do you notice?
  • Explore different triangles in relation to the pyramids!

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Discussion

Triangle Proportionality Theorem

The previous exploration can lead to discovering the following claim, which is often referred to as the Side-Splitter Theorem.

Rule

Triangle Proportionality Theorem

If a segment parallel to one of the sides of a triangle is drawn between the other sides, the segment divides the other two sides proportionally.

Proportionality Theorems for Triangles - Geometry Insights (3)

Based on the diagram, the following relation holds true.

If DEAB, then DCAD=ECBE

Proof

Since DE and AB are parallel, by the Corresponding Angles Theorem, CDE and CAB are congruent. Similarly, CED and CBA are congruent.

Proportionality Theorems for Triangles - Geometry Insights (4)

Therefore, by the Angle-Angle Similarity Theorem, ABC and DEC are similar. Consequently, their corresponding sides are proportional.

ABCDECDCAC=ECBC

Applying the Segment Addition Postulate, both numerators can be rewritten.

ACBC=AD+DC=BE+EC

Substituting these expressions into the equation above, the required proportion will be obtained.

DCAC=ECBC

SubstituteII

AC=AD+DC, BC=BE+EC

DCAD+DC=ECBE+EC

WriteSumFrac

Write as a sum of fractions

DCAD+DCDC=ECBE+ECEC

SimpQuot

Simplify quotient

DCAD+1=ECBE+1

SubEqn

LHS1=RHS1

DCAD=ECBE

Pop Quiz

Practice Using the Triangle Proportionality Theorem

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Example

Using the Triangle Proportionality Theorem to Draw Segments

In the following example the Triangle Proportionality Theorem can be used after rearranging the segments to form triangles. Given the segments on the diagram, construct a segment of length ab.

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Answer

Proportionality Theorems for Triangles - Geometry Insights (7)

Hint

Use that ab:b=a:1.

Solution

Move the slider to rearrange the given line segments. Note that this can also be done on paper using a straightedge to draw straight lines, followed by using a compass to copy the segments.

Proportionality Theorems for Triangles - Geometry Insights (8)

Label the points on this rearranged graph, connect the endpoints of the segments of length 1 and b, and draw a parallel line to this connecting line through the endpoint of the segment of length a.

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In ACE the transversal BD is parallel to the side CE. According to the Triangle Proportionality Theorem, this transversal cuts sides AC and AE proportionally.

ADDE=ABBC

Substituting the length of AB, BC, and AD gives an equation which can be solved for the length of DE.

ADDE=ABBC

SubstituteValues

Substitute values

bDE=1a

DivByOne

1a=a

bDE=a

MultEqn

LHSb=RHSb

DE=ab

This construction gave a segment of length ab.

Proportionality Theorems for Triangles - Geometry Insights (10)

Discussion

Converse Triangle Proportionality Theorem

The converse of the Side-Splitter Theorem is also true.

If a segment is drawn between two sides of a triangle such that it divides the sides proportionally, the segment is parallel to the third side in the triangle.

Proportionality Theorems for Triangles - Geometry Insights (11)

Based on the diagram, the following relation holds true.

If DCAD=ECBE, then DEAB.

Proof

The given proportion can be rearranged to get the proportionality of two sides of ABC and DEC.

DCAD=ECBE

Rewrite

AddEqn

LHS+1=RHS+1

DCAD+1=ECBE+1

OneToFrac

Rewrite 1 as aa

DCAD+DCDC=ECBE+ECEC

AddFrac

Add fractions

DCAD+DC=ECBE+EC

Segment Addition Postulate

DCAC=ECBC

This means that ABC is a dilation of DEC from point C with scale factor r=DCAC=ECBC.A dilation moves a segment to a parallel segment, so the proof is complete.

If DCAD=ECBE, then DEAB.

Example

Solving Problems With the Converse Triangle Proportionality Theorem

Show that PQRS is a parallelogram.

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Hint

Draw the diagonals of quadrilateral ABCD.

Solution

Draw diagonal AC of quadrilateral ABCD and focus on the two triangles ABC and ADC.

Proportionality Theorems for Triangles - Geometry Insights (13)

The given measures of the segments make it possible to derive the ratios according to how the transversal PQ divides sides AB and BC of ABC.

PBPAQBQC=1.84.2=37=2.45.6=37

It can be seen that the two ratios are equal. Therefore, according to the converse of the Triangle Proportionality Theorem, the transversal PQ is parallel to the diagonal AC. A similar calculation shows that SR cuts the sides of ADC proportionally. Therefore, SR is also parallel to the diagonal AC.

SDSARDRC=2.14.9=37=2.76.3=37

Since PQ and SR are both parallel to AC, they are also parallel to each other.

Proportionality Theorems for Triangles - Geometry Insights (14)

After drawing the diagonal BD, the previously found proportions also show that PS cuts the sides AB and AD of ABD proportionally. Additionally, QR cuts the sides CB and CD of CBD proportionally.

PBPAQBQC=SDSA=37=RDRC=37

This implies that PS and QR are both parallel to BD. Therefore, they are also parallel to each other.

Proportionality Theorems for Triangles - Geometry Insights (15)

This completes the proof that opposite sides of quadrilateral PQRS are parallel. Hence, by definition, it is a parallelogram.

Proportionality Theorems for Triangles - Geometry Insights (16)

Discussion

Three Parallel Lines Theorem

The following theorem is a corollary of the Side-Splitter Theorem.

If three parallel lines intersect two transversals, then they divide the transversals proportionally.

Proportionality Theorems for Triangles - Geometry Insights (17)

Applying the theorem to the diagram above, the following proportion can be written.

WYUW=XZVX

Proof

In the diagram, draw VY and let P be the point of intersection between this segment and line s.

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Next, separate UVY and YZV.

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Since WPUV, by the Triangle Proportionality Theorem WP divides YU and YV proportionally.

WYUW=PYVP

Applying the same reasoning in YZV, it can be said that PX divides VY and VZ proportionally.

PYVP=XZVX

Finally, the Transitive Property of Equality leads to the desired proportion, showing that the three parallel lines divide the transversals proportionally.

WYUW=PYVPPYVP=XZVXWYUW=XZVX

Pop Quiz

Practice Using the Three Parallel Lines Theorem

Proportionality Theorems for Triangles - Geometry Insights (20)

Example

Solving Problems Using the Three Parallel Lines Theorem

Construct points that divide the given segment into five congruent pieces.

Proportionality Theorems for Triangles - Geometry Insights (21)

Hint

Draw a different segment and extend it with four congruent copies.

Solution

Draw a ray starting at A and use a compass to copy any length five times on this ray. This gives five points, P1, P2, P3, P4, and P5.

Proportionality Theorems for Triangles - Geometry Insights (22)

Connect B with the last point, P5, and construct parallel lines to this segment through the other points. Mark the intersection points of these lines with segment AB.

Proportionality Theorems for Triangles - Geometry Insights (23)

According to the Three Parallel Lines Theorem, these transversals divide segments AB and AP5 proportionally. Since, by construction, the segments on AP5 have equal length, this means that points Q1, Q2, Q3, and Q4 divide AB into congruent segments.

Explore

Investigating Inner Triangles of a Triangle

The examples until now were based on similar triangles generated by parallel lines. Is it possible to cut a triangle to two similar triangles using a line starting from a vertex?

Move the vertices and the point on the side of the triangle. It is possible to find an arrangement when the two inner triangles are similar to each other and to the original triangle. Find such a diagram.

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Discussion

Right Triangle Similarity Theorem

The previous exploration can lead to the following claim.

Given a right triangle, if an altitude is drawn from the vertex of the right angle to the hypotenuse, then the two triangles formed are similar to the original triangle and to each other.

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According to this theorem, there are three relations that hold true for the diagram above.

  • CBDABC
  • ACDABC
  • CBDACD

Proof

Start by showing separately the two triangles formed when the altitude CD dissects ABC.

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By the Reflexive Property of Congruence, BB and AA. Also, since all right angles are congruent, it is obtained that BDCBCA and CDABCA.

CBD and ABC ACD and ABC
BB AA
BDCBCA CDABCA

Applying the Angle-Angle (AA) Similarity Theorem, it can be concluded that CBD and ABC are similar, and ACD and ABC are similar. Then, by the Transitive Property of Congruence, ACD and CBD are also similar.

CBDABC and ACDABC

CBDACD

Discussion

Corollaries of the Right Triangle Similarity Theorem

The following two claims are corollaries of the Right Triangle Similarity Theorem

Rule

Geometric Mean Altitude Theorem

Given a right triangle, if an altitude is drawn from the vertex of the right angle to the hypotenuse, then the measure of this altitude is the geometric mean between the measures of the two segments formed on the hypotenuse.

Proportionality Theorems for Triangles - Geometry Insights (27)

Based on the diagram above and by definition of the geometric mean, the following relation holds true.

CD2=ADBD or ADCD=CDBD

The Geometric Mean Altitude Theorem is also known as the Right Triangle Altitude Theorem and the Geometric Mean Theorem.

Proof

According to the Right Triangle Similarity Theorem, the two triangles formed by the altitude CD are similar.

CBDACD

Then, by definition of similar triangles, the lengths of corresponding sides are proportional.

ADCD=CDBD

Applying the Properties of Equality, this proportion can be rewritten without fractions.

CD2=ADBD

Rule

Geometric Mean Leg Theorem

Given a right triangle, if the altitude is drawn from the vertex of the right angle to the hypotenuse, then the measure of each leg of the triangle is the geometric mean between the length of the hypotenuse and the length of the segment formed on the hypotenuse adjacent to the leg.

Proportionality Theorems for Triangles - Geometry Insights (28)

Based on the diagram above, the following relations hold true.

ADAC=ACABDBCB=CBABor{AC2=ADABCB2=DBAB

Proof

According to the Right Triangle Similarity Theorem, the two triangles formed by the altitude CD are similar to ABC.

ABCACDABCCBD

This means that the corresponding parts of the triangles can be identified.

ABCACD ABCCBD
ABandAC

ACandAD
CBandCD

ABandCB

ACandCD
CBandDB

Then, by definition of similar triangles, the lengths of corresponding sides are proportional.

ABCACD ABCCBD
ADAC=ACAB DBCB=CBAB

Note that for any two pairs of corresponding sides a similar proportion can be obtained. Now, applying the Properties of Equality, the proportion can be rewritten without fractions.

{AC2=ADABCB2=DBAB

Example

Solving Problems With the Geometric Mean Leg Theorem

Represented on the figure ABC is a right triangle and an altitude CD.

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Using the given measurements, find the length of CD.

Hint

First, find the length of DB.

Solution

The answer can be found in two steps. The first step is to find the length of DB.

Finding DB

The Geometric Mean Leg Theorem shows the connection between the length of AB, DB, and CB.

CB2=DBAB

The length of AB can be expressed using the length of DB.

Proportionality Theorems for Triangles - Geometry Insights (30)

By substituting these expressions in the equation given by the Geometric Mean Leg Theorem, the results can be expressed in an equation that can then be solved for the length of DB.

CB2=DBAB

SubstituteExpressions

Substitute expressions

132=x(28.8+x)

Rewrite

Distr

Distribute x

132=28.8x+x2

SubEqn

LHS132=RHS132

0=28.8x+x2132

CalcPow

Calculate power

0=28.8x+x2169

RearrangeEqn

Rearrange equation

x2+28.8x169=0

UseQuadForm

Use the Quadratic Formula: a=1,b=28.8,c=-169

x=2(1)-28.8±28.824(1)(-169)

Evaluate right-hand side

CalcPowProd

Calculate power and product

x=2-28.8±829.44+676

AddTerms

Add terms

x=2-28.8±1505.44

CalcRoot

Calculate root

x=2-28.8±38.8

WriteSumFrac

Write as a sum of fractions

x=-228.8±238.8

CalcQuot

Calculate quotient

x=-14.4±19.4

This gives two solutions, x=-14.4+19.4=5, and x=-14.419.4=-33.8. Since x represents the length of segment DB, only the positive solution is meaningful, as a length of a segment can not be negative.

DB=5cm

Finding CD

This gives the length of the other segment formed by the altitude in the right triangle ABC.

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The Geometric Mean Altitude Theorem gives a connection between the length of AD, DB, and CD.

CD2=ADDB

This relationship can be used to find the length of the altitude.

CD2=ADDB

SubstituteII

AD=28.8, DB=5

CD2=28.8(5)

Solve for CD

Multiply

Multiply

CD2=144

CalcRoot

Calculate root

CD=12

The length of the altitude CD is 12 centimeters.

Pop Quiz

Practice the Geometric Mean Leg Theorem

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Discussion

Pythagorean Theorem

For right triangles, the length of the hypotenuse squared equals the sum of the squares of the lengths of the legs.

Proportionality Theorems for Triangles - Geometry Insights (33)

Proof

Using Similarity

First, draw the altitude from the right angle to the hypotenuse. This divides the hypotenuse into two segments.

Proportionality Theorems for Triangles - Geometry Insights (34)

Next, apply the Geometric Mean Leg Theorem. Doing this relates the lengths of the legs to the length of the hypotenuse.

{a2=xcb2=yc

These two equations can be added. Then, c can be factored out from the right-hand side.

a2+b2a2+b2a2+b2=xc=yc=xc+yc=(x+y)c

Drawing the altitude resulted in the outcome that x+y is equal to c. After substituting this into the equation above and simplifying, the Pythagorean Theorem is obtained.

a2+b2a2+b2=cc=c2

Closure

Solving Problems Using Triangle Similarity

To conclude this lesson, the opening challenge will be revisited. The challenge shows a diagram consisting of the Grim Reaper and two street lamps at 15 and 10 feet tall. How tall is the Grim Reaper?

Hint

The lamps, the head of the Grim Reaper, and the shadows of the Grim Reaper's head are on a straight line.

Solution

The lamps and the figure stand vertically. Hence, they can be represented by parallel segments AB, CD, and EF. Notice that the shadows just reach the lampposts. Taking a look at the shadow touching the taller post, it can be derived that the lamppost bottom B, the head of the Grim Reaper C, and the lamppost top E are on a straight line. Applying this same logic to the other shadow implies that A, C, and F are also on a straight line.

Proportionality Theorems for Triangles - Geometry Insights (36)

Segment CD splits both ABF and BEF. It is also parallel to one side of both triangles. That means the combination of two dilations maps AB to EF.

  • A dilation with center F and scale factor DF:BF maps AB to CD.
  • A dilation with center B and scale factor BF:BD maps CD to EF.

The scale factor of this combined similarity transformation is the product of the two scale factors.

Proportionality Theorems for Triangles - Geometry Insights (37)

Continuing, notice that Point D is between B and F. Therefore, the Segment Addition Postulate guarantees that BD+DF=BF. Then, to divide this equality by BF and to rearrange it gives a relationship between the scale factors of the two dilations.

BD+DF=BF

Rewrite

DivEqn

LHS/BF=RHS/BF

BFBD+BFDF=1

SubEqn

LHSBFDF=RHSBFDF

BFBD=1BFDF

LHS1=RHS1

BDBF=1BFDF1

The product of the two scale factors is the scale factor of the similarity transformation that maps the 15-foot lamppost to the 10-foot lamppost. Hence, the scale factor is 10:15. An equation can now be written and solved to find the individual scale factors.

BFDFBDBF=1510

SubstituteExpressions

Substitute expressions

BFDF1BFDF1=1510

Solve for BFDF

Substitute

BFDF=x

x1x1=1510

MultEqn

LHS15(1x)=RHS15(1x)

15x=10(1x)

Distr

Distribute 10

15x=1010x

AddEqn

LHS+10x=RHS+10x

25x=10

DivEqn

LHS/25=RHS/25

x=2510

CalcQuot

Calculate quotient

x=0.4

Substitute

x=BFDF

BFDF=0.4

This finding, 0.4, is the scale factor of the dilation that maps AB to CD. Since AB=15, the length of CD can now be calculated.

CD=0.4(15)=6

Segment CD represents the Grim Reaper, who is now known to stand at 6 feet tall.

Proportionality Theorems for Triangles - Geometry Insights (2024)

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