Subject Code: Math 4 Mathematics 4
Module Code: 7.0 Triangle Similarity
Lesson Code: 7.2.1 The Basic Proportionality Theorem Part 1
Time Limit: 30 minutes
Resource(s) graphing application (preferably GeoGebra)
By the end of this lesson, the student will have been able to apply the Basic Proportionality Theorem in
solving the missing parts of a triangle.
Introduction (Time Allotment: 5 minutes)
Triangles are the simplest polygons having the least number of sides, yet they also possess many
interesting properties involving their sides and interior angles. In this lesson, you will learn yet another
property of triangles that has something to do with the lengths of its sides.
Let’s begin with a puzzle involving triangles. How many triangles can you see in the figure below? 12?
16? 18? 22?
Exploring Proportionality in a Triangle (Time Allotment: 7 minutes)
In this activity, you will need a graphing application (preferably GeoGebra).
Open your GeoGebra application and do the following:
1. Select Option > Labeling > New Points Only (For mobile device: Settings > General >
Labeling > New Points Only)
2. Hide the Axes and Grids from the Graphics window: Click the Graphics toggle, then click the
Axes and Grids icons (for mobile device: Settings > Graphics, then turn off the Show Axes and
Show Grid buttons)
Step 1: Make 4ABC.
Select the Polygon tool and click three points on the graphics window. Make sure to go back to
the first point created. A triangle with vertices A, B and C appears.
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Step 2: Plot a point on one side of 4ABC.
Select the Point tool. Choose a position on the side AB randomly and click on it. Point D appears.
Step 3: Make a line parallel to side BC through point D.
Select the Parallel Line tool, then in succession, click point D and side BC. A line appears.
This line is parallel to BC and passes through point D.
Step 4: Locate the point of intersection of side AC and the line.
Select the Intersect tool, then in succession click side AC and the line parallel to BC. Point E
appears. Below is a sample output after this step.
Step 5: Measure the lengths of the cut segments
Select the Distance or Length tool, then in succession, click points A and D – the length of
segment AD appears. Next, click points D and B - the length of segment DB appears. Next, click
points A and E - the length of segments AE appears. Lastly, click points E and C - the length of
segment EC appears.
Step 6: Compute the ratios
AD AE
With your calculator, compute the ratios and . What do you observe?
DB EC
←→
Step 7: Measure the lengths of the sides of 4ABC intersected by DE.
Select the Distance or Length tool, then in succession, click points A and B - the length of side
AB appears. Next, click points A and C - the length of side AC appears.
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Step 8: Compute the ratios.
AD AE
Compute the ratios and .What do you observe?
AB AC
DB EC
Now compute and . What do you observe?
AB AC
AD AE
When the above steps are followed correctly, you should be able to see that the ratios and are
DB EC
AD AE DB EC
equal (Step 6). Same is also true for ratios and , as well as for and (Step 7). If you move
AB AC AB AC
one vertex of 4ABC to another position, and recompute the above ratios, you will see that the equality
remains.
This property of triangles is stated below as a theorem.
The Basic Proportionality Theorem (Time Allotment: 3 minutes)
The Basic Proportionality Theorem
If a line intersects any two sides of any triangle and is parallel to the third side, then that line
divides the two intersected sides of the triangle proportionally.
Restatement of the Basic Proportionality Theorem
←
→
If, in 4ABC (shown below), line XY intersects the sides AB and AC at points X and Y respectively, and
AX AY
is parallel to side BC, then = .
XB YC
AB AC AB AC
The following proportions also follows from the theorem: = and = .
AX AY XB YC
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Sample Problems (Time Allotment: 10 minutes)
Example 1
Given the figure below (not drawn to scale) with PT parallel to MO, answer the questions that follow.
a) If OT = 3, T G = 7 and MP = 5, what is the length of PG?
b) If MG = 12, PG = 9 and OT = 6, what is the length of T G?
c) If MP = x, PG = x + 4, OT = x − 1 and T G = (7x + 1) /x, what is the length of MG?
Solution.
a) By the Basic Proportionality Theorem,
OT MP
=
TG PG
Thus,
3 5
=
7 PG
Solving for PG, we get
35
PG =
3
b) By the Basic Proportionality Theorem,
MG OG
=
PG TG
Substituting the values of MG and PG, and replacing OG with OT + T G, we then have
12 OT + T G
=
9 TG
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4 6+TG
=
3 TG
4(T G) = 3(6 + T G) = 18 + 3T G
4T G − 3T G = 18
T G = 18
c) By the Basic Proportionality Theorem,
PG TG
=
MP OT
Substituting the corresponding lengths, we then have
x+4 (7x + 1)/x
=
x x−1
x+4 7x + 1 1
=
x x x−1
x+4 7x + 1
=
x x(x − 1)
x(x − 1)(x + 4) = x(7x + 1)
x3 + 3x2 − 4x = 7x2 + x
x3 − 4x2 − 5x = 0
x(x2 − 4x − 5) = 0
x(x − 5)(x + 1) = 0
Hence, the possible values of x are 0, 5 or −1. When x is either 0 or −1, the length of MP becomes
0 or negative. This makes 0 or −1 an invalid value for x. When x = 5, MP = 5, PG = 9, OT = 4
and T G = 36/5 which are all positive values. Moreover,
PG 9
=
MP 5
and
T G 36/5 36 9
= = =
OT 4 20 5
Hence, x = 5. Therefore, MG = MP + PG = 5 + 9 = 14.
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Example 2
Given the figure below with ∠AXY ∼
= ∠ABC, find the length of YC.
Solution.
Segment XY intersects the two sides of 4ABC at X and Y , respectively. The Basic Proportionality
Theorem can only be applied if we can establish that XY is parallel to BC.
←→ ←→
Imagine in the figure that there are lines XY and BC that contain segments XY and BC, respectively. It
←
→ ←→
can then be seen that ∠AXY and ∠ABC are corresponding angles formed by lines XY and BC cut by
←→
transversal AB. These angles are given to be congruent, hence, we can conclude that the lines XY and
←→ ←→ ←→
BC are parallel (Why?). Since segments XY and BC are contained in lines XY and BC, respectively, it
then follows that XY is parallel to BC. With this we can say that the Basic Proportionality Theorem can
be applied.
Applying the Basic Proportionality Theorem, we get
AX AY
=
XB YC
which implies that
2 3
=
6 YC
2(YC) = 3(6)
YC = 9
Thus, as required, the length of YC is as computed above.
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Corollary to the Triangle Sum Theorem (Time Allotment: 5 minutes)
There is a very important principle which can be gleaned from the last example given. Please read on to
learn such principle which you will use when you study similarities in triangles (Lesson 7.3).
The figure below shows the two triangles 4AXY and 4ABC from the figure in Example 2.
Note that the two triangles share a common angle, ∠A. Thus, we can say that the two triangles have two
pairs of corresponding angles which are given to be congruent. What can be said about the third pair
of corresponding angles of the triangles above? To answer this question, consider, in general, any two
triangles 4ABC and 4DEF such that ∠A ∼ = ∠D and ∠B ∼ = ∠E (See figure below). What could be true
about ∠C and ∠F?
At this point, recall that the sum of the measures of the interior angles of any triangle is 180 (Triangle
Sum Theorem). Applying this theorem, we get
m∠A + m∠B + m∠C = 180
and
m∠D + m∠E + m∠F = 180
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By the Transitive Property of Equality,
m∠A + m∠B + m∠C = m∠D + m∠E + m∠F
Since ∠A ∼= ∠D and ∠B ∼= ∠E, then m∠A = m∠D and m∠B = m∠E by the definition of congruent
angles. Replacing m∠A and m∠B with m∠D and m∠E, respectively in the last equation, we get
m∠D + m∠E + m∠C = m∠D + m∠E + m∠F
This implies that m∠C = m∠F. The third pair of angles of the triangles are then shown to be congruent.
The generalization of this result is given below as a corollary to the Triangle Sum Theorem.
Corollary to Triangle Sum Theorem
If two triangles have two pairs of congruent corresponding angles, then the third pair of
corresponding angles are also congruent.
For the next lesson, you shall use the Basic Proportionality Theorem in solving for the missing side or
sides of triangles.
References
Albarico, J.M. (2013). THINK Framework. Based on Ramos, E.G. and Apolinario, N. (n.d.). Science
LINKS. Quezon City: Rex Bookstore Inc.
Daniels, A. (2020). We Spent All Day Arguing About This Triangle Brain Teaser. Can You Solve It?
Retrieved from https://www.popularmechanics.com/science/math/a30706968/viral-triangle-brain
-teaser-solved/
Hohenwarter, M., Borcherds, M., Ancsin, G., Bencze, B., Blossier, M., Éliás, J., . . . Tomaschko, M.
(2018, October). GeoGebra 5.0.507.0. (http://www.geogebra.org)
Moise, E. E., & Downs, F. L., Jr. (1991). Geometry. Addison-Wesley Publishing Company.
Prepared by: Mr. Harold C. Soriano Reviewed by: Mary Gay Antonette G. Magpantay
Position: Special Science Teacher IV Position: Special Science Teacher III
Campus: PSHS-SMC Campus: PSHS-MC
c 2020 Philippine Science High School System. All rights reserved. This document may contain proprietary
information and may only be released to third parties with approval of management. Document is uncontrolled
unless otherwise marked; uncontrolled documents are not subject to update notification.
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