In this explainer, we will learn how to find missing lengths in a triangle containing two or three parallel lines using proportionality. Recall that when two parallel lines are cut by a transversal, the resulting corresponding angles are equal. By adding a second transversal as pictured below, we can form two triangles. Giving each vertex a label, we can define the larger triangle β³π΄π·πΈ and the smaller triangle β³π΄π΅πΆ. Since the two pairs of corresponding angles are equal, triangle π΄π·πΈ is similar to triangle π΄π΅πΆ: β³π΄π·πΈβΌβ³π΄π΅πΆ. Since these triangles are similar, the ratios of their corresponding side lengths must be equal. In other words, we have π΄π΅π΄π·=π΄πΆπ΄πΈ=π΅πΆπ·πΈ. In the first example, we will demonstrate how to use this definition of the similarity of triangles to identify which pairs of side lengths have equal proportions when a triangle is cut by a line parallel to one of its sides. Using the diagram, which of the following is equal to π΄π΅π΄π·? The diagram indicates that πΈπ· is parallel to πΆπ΅. Since corresponding angles are equal, that is, β π·πΈπ΄=β π΅πΆπ΄ and β πΈπ·π΄=β πΆπ΅π΄,πΈπ· creates triangle π΄π·πΈ that is similar to the larger triangle π΄π΅πΆ. Since these triangles are similar, the ratios of their corresponding side lengths must be equal. In particular, π΄πΈπ΄πΆ=π΄π·π΄π΅. To find the fraction that is equivalent to π΄π΅π΄π·, we can find the reciprocal of both sides of this equation: π΄πΆπ΄πΈ=π΄π΅π΄π·. π΄πΆπ΄πΈ is equal to π΄π΅π΄π·. Find the value of π₯. ο«π΄πΆ and ο«π΄π΅ are transversals that intersect parallel lines βο©ο©ο©ο©βπ·πΈ and βο©ο©ο©ο©βπ΅πΆ. Since the two pairs of corresponding angles created by this intersection are equal, that is, β π·πΈπ΄=β π΅πΆπ΄,β πΈπ·π΄=β πΆπ΅π΄, we can say that triangle π΄π·πΈ is similar to triangle π΄π΅πΆ: β³π΄π΅πΆβΌβ³π΄π·πΈ. When two triangles are similar, the ratios of the lengths of their corresponding sides are equal. In particular, π΄π·π΄π΅=π·πΈπ΅πΆ. By substituting in known values for the lengths of π΄π·, π·πΈ, and π΄π΅ (where we should note that π΄π΅ is the sum of π΄π· and π·π΅), we can find the value of π₯: 1010+11=10π₯. Solving for π₯, π₯=21.Answer
Answer
In the previous two examples, we noted that, if a line intersecting two sides of a triangle is parallel to the third side, then the smaller triangle created by the parallel line is similar to the original triangle. We recall the diagram we presented earlier.
Since triangles π΄π΅πΆ and π΄π·πΈ are similar, we obtain the equal proportions: π΄π΅π΄π·=π΄πΆπ΄πΈ.
From this diagram, we also note that the line segments π΄π· and π΄πΈ can be split as follows: π΄π·=π΄π΅+π΅π·π΄πΈ=π΄πΆ+πΆπΈ.and
Substituting these expressions into our earlier equation and rearranging, π΄π΅π΄π·=π΄πΆπ΄πΈπ΄π΅π΄π΅+π΅π·=π΄πΆπ΄πΆ+πΆπΈπ΄π΅(π΄πΆ+πΆπΈ)=π΄πΆ(π΄π΅+π΅π·)π΄π΅β π΄πΆ+π΄π΅β πΆπΈ=π΄πΆβ π΄π΅+π΄πΆβ π΅π·.
We can now subtract π΄π΅β π΄πΆ from both sides to find π΄π΅β πΆπΈ=π΄πΆβ π΅π·,π΄π΅π΅π·=π΄πΆπΆπΈ.
This leads us to the definition of a theorem that links the line segments created when a parallel side is added to a triangle.
If a line parallel to one side of a triangle intersects the other two sides of the triangle, then the line divides those sides proportionally.
Note:
The side splitter theorem can be extended to include parallel lines that lie outside of the triangle. When a straight line lies outside of a triangle and is parallel to one side of the triangle, it forms another triangle that is similar to the first one. This is demonstrated in the following diagram. In this case, an analog of the side splitter theorem can be deduced directly from the similar triangles.
In our next example, we will see how to use this theorem to identify proportional segments of triangles to calculate a missing length.
In the figure, ππ and π΅πΆ are parallel. If π΄π=18, ππ΅=24, and π΄π=27, what is the length of ππΆ?
Answer
We are given that ππ is parallel to π΅πΆ. The side splitter theorem says that if a line parallel to one side of a triangle intersects the other two sides of the triangle, then the line divides those sides proportionally.
In particular, π΄πππΆ=π΄πππ΅.
Substituting π΄π=18, ππ΅=24, and π΄π=27 into this equation and solving for ππΆ, 27ππΆ=1824ππΆ27=2418ππΆ=2418Γ27=36.
The length of ππΆ is 36.
In our next example, we will demonstrate how to solve multistep problems involving triangles and parallel lines.
The given figure shows a triangle π΄π΅πΆ.
- Work out the value of π₯.
- Work out the value of π¦.
Answer
Part 1
In the figure, a line parallel to side π΅πΆ is intersecting the other two sides of the triangle. The side splitter theorem tells us that this line divides those sides proportionally.
Labelling this line segment as π·πΈ, we obtain π΄π·π·π΅=π΄πΈπΈπΆ.
This gives us an equation that can be solved for π₯: 32π₯+3=2π₯+53(π₯+5)=2(2π₯+3)3π₯+15=4π₯+615=π₯+6π₯=9.
Part 2
Now that we know the value of π₯, we can use this information to find the value of π¦. Since the two pairs of corresponding angles created by the intersection of π·πΈ are equal, triangle π΄π΅πΆ is similar to triangle π΄π·πΈ: β³π΄π΅πΆβΌβ³π΄π·πΈ.
In particular, π΄π·π΄π΅=π·πΈπ΅πΆ.
The length of π΄π΅ is the sum of the lengths of π΄π· and π·π΅. We are given that π΄π·=3 and π·π΅=2π₯+3. Since π₯=9, π·π΅=21. Therefore, π΄π΅=3+21=24.
Substituting these values into our earlier equation and solving for π¦, 324=2π¦π¦24=23π¦=23Γ24=16.
Therefore, π¦=16.
In our next example, we will demonstrate how to apply the side splitter theorem to a triangle which contains several pairs of parallel lines.
Find the length of πΆπ΅.
Answer
From the given diagram we note that π·πΉ is parallel to π΄πΈ in the triangle πΆπ΄πΈ, and π·πΈ is parallel to π΄π΅ in the triangle πΆπ΄π΅. The side splitter theorem tells us that if a line parallel to one side of a triangle intersects the other two sides of the triangle, then the line divides those sides proportionally. Applying this theorem to triangle πΆπ΄πΈ where π·πΉ is parallel to one side of the triangle, we obtain πΆπΉπΉπΈ=πΆπ·π·π΄.
Since π·πΈ is parallel to one side of the larger triangle πΆπ΄π΅, we can also obtain πΆπΈπΈπ΅=πΆπ·π·π΄.
Both πΆπΉπΉπΈ and πΆπΈπΈπ΅ are equal to πΆπ·π·π΄. This means we can set πΆπΉπΉπΈ=πΆπΈπΈπ΅.
We can substitute the given values πΆπΉ=15, πΉπΈ=6, and πΆπΈ=15+6=21 into this equation to obtain an equation that can be solved for πΈπ΅: 156=21πΈπ΅πΈπ΅=21Γ615.
Therefore, πΈπ΅=8.4.cm
Since πΆπ΅=πΆπΉ+πΉπΈ+πΈπ΅, πΆπ΅=15+6+8.4=29.4.cm
The length of πΆπ΅ is 29.4 cm.
Recall that the side splitter theorem tells us that if a line parallel to one side of a triangle intersects the other two sides of the triangle, then the line divides those sides proportionally. Moreover, we have learned that this theorem can be extended to include parallel lines that lie outside of the triangle. It turns out that the converse of this result is also true, which proves very useful when solving problems of this type.
If a line intersects two sides of a triangle and splits those sides in equal proportions, then that line must be parallel to the third side of the triangle.
In all three diagrams above, π΄π΅πΆ is a triangle and βο©ο©ο©ο©βπ·πΈ intersects βο©ο©ο©ο©βπ΄π΅ at π· and βο©ο©ο©ο©βπ΄πΆ at πΈ.
If π΄π·π·π΅=π΄πΈπΈπΆ, then βο©ο©ο©ο©βπ·πΈ must be parallel to βο©ο©ο©ο©βπ΅πΆ.
By applying the converse of the side splitter theorem, we are able to prove that a straight line is parallel to one side of a triangle due to having proportional parts. In our final example, we will demonstrate this process.
Given that π΄π΅πΆπ· is a parallelogram, find the length of ππ.
Answer
To find the length of ππ, we will begin by identifying relevant information about triangles πππ and ππ·πΆ. We are given that ππ=ππ· and ππ=ππΆ. We also recall that the side splitter theorem tells us that if a line parallel to one side of a triangle intersects the other two sides of the triangle, then the line divides those sides proportionally. Conversely, if a line splits two sides of a triangle into equal proportions, then that line must be parallel to the third side. Since sides ππ· and ππΆ of the larger triangle ππ·πΆ have been divided into equal proportions, we can apply the converse of this theorem to deduce that π·πΆ and ππ must be parallel.
We also recall that if a line parallel to a side of a triangle intersects two other sides, then the smaller triangle created by the parallel line is similar to the original triangle. Hence, we obtain β³πππβΌβ³ππ·πΆ.
Since π·πΆ is the opposite side of π΄π΅ in the parallelogram π΄π΅πΆπ·, these two sides must have the same lengths. Hence, the length of π·πΆ is 134.9 cm. Denoting the length of ππby an unknown constant π₯, we can draw the following diagram.
Since triangles πππ and ππ·πΆ are similar, we can form an equation that links the lengths of the sides ππ, ππ·, ππ, and π·πΆ: ππππ·=πππ·πΆπ₯2π₯=ππ134.912=ππ134.9.
Solving for ππ, we find ππ=134.92=67.45.
The length of ππ is 67.45 cm.
We will now recap the key points from this explainer.
- If a line intersecting two sides of a triangle is parallel to the remaining side, then the smaller triangle created by the parallel line is similar to the larger, original triangle.
- The side splitter theorem tells us that if a line parallel to one side of a triangle intersects the other two sides of the triangle, then the line divides those sides proportionally.
- The side splitter theorem can be extended to include parallel lines that lie outside a triangle. If a line lying outside a triangle is parallel to one side of the triangle and intersects the extensions of the other two sides of the triangle, then the line divides the extensions of those sides proportionally.
- The converse of the side splitter theorem states that if a line splits two sides of a triangle proportionally, then that line is parallel to the remaining side.