Triangle Proportionality Theorem: Proof of the Theorem & Sample Questions

Triangle Proportionality Theorem is a fundamental concept that establishes a relationship between the sides of a triangle. Also known as the "Side Splitter Theorem" or the "Transversal Theorem," this theorem plays a crucial role in understanding the proportional relationships that exist within triangles. 

• The Triangle Proportionality Theorem states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally. 
• This means that the lengths of the segments formed on the two sides of the triangle are in proportion to each other. 
• In other words, when a line intersects two sides of a triangle, forming several segments, the lengths of these segments will be related in a predictable way. 
• This theorem provides a precise mathematical explanation for this relationship and allows us to make accurate calculations based on the given information.

Key Terms: Triangle, Similarity, Proportional, Parallel, Side, Angle, Intersection, Ratio.

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What is Triangle Proportionality Theorem?

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The triangle proportionality theorem states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.

Triangle Proportionality Theorem

Considering a triangle ABC and drawing a line parallel to one of its sides, such as side BC, intersecting the other two sides at points P and Q, the theorem states that the resulting segments created on sides AB and AC, namely AP and AQ, are proportional to the corresponding segments on side BC, namely BP, and CQ.

Mathematically, the Triangle Proportionality Theorem can be expressed as

AP/PC = AB/BC = AQ/QB

This means that the ratios of the lengths of the corresponding segments are equal. It is important to note that this theorem applies not only to the lengths of the segments but also to their directed lengths, taking into account their positive or negative direction.

Read More: Similarity of Triangles

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Proof of the Triangle Proportionality Theorem

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To prove the Triangle Proportionality Theorem, start with a triangle ABC and draw a line parallel to side BC, intersecting sides AB and AC at points P and Q, respectively.

To demonstrate that the resulting segments AP, AQ, BP, and CQ are proportional, the concept of similar triangles is used. Two triangles are similar if their corresponding angles are congruent and the lengths of their corresponding sides are proportional.

First, consider triangle ABC and triangle AQP. Since line PQ is parallel to side BC, the following pairs of corresponding angles are found:

∠BAP ≅ ∠AQP (corresponding angles)

∠BCA ≅ ∠AQP (corresponding angles)

This establishes that triangle ABC and triangle AQP are similar by the Angle-Angle (AA) criterion for triangle similarity.

Next, comparing the corresponding sides of the two similar triangles:

AB/AQ = AC/AP (corresponding sides of similar triangles)

Similarly, let's consider triangle ABC and triangle BPQ. Again, using the fact that line PQ is parallel to side BC:

∠CBA ≅ ∠BQP (corresponding angles)

∠BCA ≅ ∠QBP (corresponding angles)

This indicates that triangle ABC and triangle BPQ are similar by the Angle-Angle (AA) criterion.

Now, let's compare the corresponding sides of these similar triangles:

AB/BP = AC/CQ (corresponding sides of similar triangles)

Combining the two proportional relationships obtained from the similar triangles, the obtained equation is:

AB/AQ = AC/AP = AB/BP = AC/CQ

This demonstrates that the segments AP, AQ, BP, and CQ formed by the line parallel to side BC are proportional.

Read More: Questions on Triangles

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Converse of the Triangle Proportionality Theorem

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The Converse of the Triangle Proportionality Theorem establishes a relationship between the sides of a triangle when certain proportions are present. It is essentially the opposite statement of the Triangle Proportionality Theorem.

The Converse of the Triangle Proportionality Theorem states that if a line divides two sides of a triangle proportionally, then the line must be parallel to the third side of the triangle.

Converse of the Triangle Proportionality Theorem

Consider a triangle ABC. Suppose there is a line that intersects sides AB and AC at points P and Q, respectively, such that the resulting segments AP, AQ, and PQ satisfy the following proportion:

AP/PB = AQ/QC

The Converse of the Triangle Proportionality Theorem asserts that in this scenario, the line PQ must be parallel to the third side, BC.

Read More: Triangles Important Question

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Proof of the Converse Triangle Proportionality Theorem

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To prove the Converse of the Triangle Proportionality Theorem, the concept of similar triangles is utilized. If the similarity of certain triangles is established, then the parallelism of the lines can be deduced.

Since the ratio AP/PB = AQ/QC, we can rewrite this ratio as

AP/PB = AQ/QC = AP/AQ

Notice that the last equality is obtained by cross-multiplying the first two ratios. Simplifying further:

PB = QC

This implies that the lengths of segments PB and QC are equal.

Now, considering triangles APB and AQC:

∠APB = ∠AQC (corresponding angles)

∠ABP = ∠ACQ (corresponding angles)

PB = QC (established from the previous step)

By the Angle-Angle (AA) criterion for similarity, it is seen that the triangles APB and AQC are similar.

Since these two triangles are similar, their corresponding sides must be proportional. This leads to the conclusion:

AB/AP = AC/AQ

By comparing this proportion to the original proportion AP/PB = AQ/QC, it is found that the ratios are equal. Therefore:

AB/AP = AC/AQ = AP/PB = AQ/QC

This indicates that the line PQ is parallel to the third side BC, as required by the Converse of the Triangle Proportionality Theorem.

Read More: Trigonometry Formulae

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Solved Example

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Example 1: Consider a triangle ABC, where AB = 6 cm, AC = 8 cm, and BC = 10 cm. A line is drawn parallel to side BC, intersecting sides AB and AC at points P and Q, respectively. We are given that AP = 2 cm and PQ = 4 cm. Find the length of AQ.

Solution: According to the theorem, the following proportion is:

AP/PC = AB/BC = AQ/QB

Substituting the given values, the given equation will be:

2/PC = 6/10 = AQ/QB

As previously established, PC = BC - BP, so substituting the values:

2/(BC - BP) = 6/10

Cross-multiplying, the obtained equation will be:

10 * 2 = 6 * (BC - BP)

20 = 6BC - 6BP

Rearranging the equation:

6BP = 6BC - 20

Dividing both sides by 6:

BP = BC - 20/6

Simplifying further:

BP = BC - 10/3

Now, since BP + PQ = BQ, the values can be substituted:

BC - 10/3 + 4 = BQ

BC - 10/3 + 12/3 = BQ

BC + 2/3 = BQ

Now, as BC = 10 cm, so substituting the value, it is found that:

10 + 2/3 = BQ

10 + (2/3) = (30/3) + (2/3) = 32/3

Therefore, BQ = 32/3 cm.

Hence, the length of AQ is equal to the length of BQ, so AQ = BQ = 32/3 cm.

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Things to Remember

• The Triangle Proportionality Theorem establishes a relationship between the sides of a triangle when a line is drawn parallel to one side.
• The theorem states that the line divides the other two sides proportionally.
• It is also known as the Side Splitter Theorem or the Transversal Theorem.
• The theorem can be expressed as AP/PC = AB/BC = AQ/QB, where AP, PC, AB, BC, AQ, and QB are the segments formed by the parallel line.
• The theorem can be proved using the concept of similar triangles.
• The Triangle Proportionality Theorem is the basis for solving geometric problems involving triangles.
• It enables us to determine unknown side lengths, find missing angles, and establish proportional relationships between different segments.