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In this explainer, we will learn how to use the triangle midsegment theorem to prove the parallelism of lines in a triangle or find a missing side length. Letβs begin with understanding what the triangle midsegment theorem states. The line segment passing through the midpoint of one side of a triangle that is also parallel to another side of the triangle bisects the third side of the triangle. We can prove this by considering triangle π΄πΆπ· with line segment πΈπ΅ that passes through the midpoint, πΈ, of π΄π· and that is parallel to π·πΆ. We can construct line βο©ο©ο©ο©ο©βπ΄π such that βο©ο©ο©ο©ο©βπ΄πβ«½πΈπ΅β«½π·πΆ. Line segments π΄π· and π΄πΆ are transversals of these three parallel lines. We recall that if a set of parallel lines divides a transversal into segments of equal lengths, then that set divides any other transversal into segments of equal lengths. Given that π΄π· was split into two congruent segments by the parallel lines, βο©ο©ο©ο©ο©βπ΄π, πΈπ΅, and π·πΆ, then the other transversal, π΄πΆ, must also be split into two congruent segments. Hence, π΄π΅=π΅πΆ. Therefore, we have proven that the third side of the triangle has been bisected. The converse of this theorem is also true, that is, if we have a triangle where two sides are bisected by a line segment, then that line segment is parallel to the third side. This is defined below. The line segment joining the midpoints of two sides of a triangle is parallel to the third side. We can also see one more of the triangle midsegment theorems. The length of the line segment joining the midpoints of two sides of a triangle is equal to half the length of the third side. Letβs examine how we can prove this theorem. Consider β³π΄πΆπ·, where πΈ and π΅ are the midpoints of π΄π· and π΄πΆ, respectively, such that π΄πΈ=πΈπ· and π΄π΅=π΅πΆ. We can construct ray οͺπΈπΉ such that πΈπΉβ«½π΄πΆ and οͺπΈπΉ intersects π·πΆ at πΉ. We can then apply the triangle midsegment theorem, which states that the line segment joining the midpoints of two sides of a triangle is parallel to the third side. This means that πΈπ΅β«½π·πΆ. As we constructed πΈπΉβ«½π΄πΆ, then by the triangle midsegment theorem, which states that the line segment passing through the midpoint of one side of a triangle that is also parallel to another side of the triangle bisects the third side of the triangle, we have that π·πΆ is bisected. Hence, π·πΉ=πΉπΆ, and πΉ is the midpoint of π·πΆ. Furthermore, πΉπΆ=12π·πΆ. We can then consider quadrilateral πΈπ΅πΆπΉ. πΈπ΅πΆπΉ is a quadrilateral with two pairs of opposite sides parallel, which by definition is a parallelogram. In a parallelogram, opposite sides are congruent. Hence, πΈπ΅=πΉπΆ=12π·πΆ. Therefore, we have proven the theorem: the line segment joining the midpoints of two sides of a triangle is equal to half the length of the third side. It is worth noting that theorems 1 and 2 here are often referred to collectively as the triangle midsegment theorem, stated as follows: the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half its length. In the following questions, we will see how we can apply the triangle midsegment theorem and its converse in order to find unknown side lengths, beginning with a question where we need to find the perimeter of a shape. Given that π· and πΈ are the midpoints of π΄π΅ and π΄πΆ, respectively, π΄π·=32cm, π΄πΈ=19cm, and π·πΈ=39cm, determine the perimeter of π·π΅πΆπΈ. We can begin by filling in the given length information on the figure, which is that π΄π·=32cm, π΄πΈ=19cm, and π·πΈ=39cm. As π· and πΈ are the midpoints of π΄π΅ and π΄πΆ, we know that πΈπΆ=π΄πΈ=19cm and π·π΅=π΄π·=32.cm The triangle midsegment theorem states that the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half its length. Thus, we can say that πΆπ΅β«½π·πΈ and πΆπ΅=2Γ(π·πΈ). Given that π·πΈ=39cm, we have πΆπ΅=2Γ39=78.cm Finally, we need to calculate the perimeter of π·π΅πΆπΈ. This is the distance around the outside of π·π΅πΆπΈ, the trapezoid on the lower part of the figure. Using π·πΈ=39cm, πΈπΆ=19cm, πΆπ΅=78cm, and π·π΅=32cm, we have perimeterofcmπ·π΅πΆπΈ=π·πΈ+πΈπΆ+πΆπ΅+π·π΅=39+19+78+32=168. Thus we have the answer that the perimeter of π·π΅πΆπΈ is 168 cm. In the next example, we will apply the triangle midsegment theorem a number of times in different triangles in the same figure. When working through a problem such as this, it can be helpful to outline or highlight the specific triangles so that we can correctly identify the key segments that we are working with. In the figure shown, πΈ, πΉ, and π· are the midpoints of π΅πΆ, π΄π΅, and π΄πΆ respectively. Find the perimeter of β³πΈπΉπ·. We are given the information that πΈ, πΉ, and π· are the midpoints of π΅πΆ, π΄π΅, and π΄πΆ respectively. In order to calculate the perimeter of β³πΈπΉπ·, we will need to determine the lengths of πΉπ·, π·πΈ, and πΈπΉ. To do this, we can recall that the length of the line segment joining the midpoints of two sides of a triangle is equal to half the length of the third side. Letβs consider πΉπ·. πΉπ· is a line segment connecting the midpoints of two sides of a triangle. Thus, πΉπ· is half the length of π΅πΆ (4.6 cm). Therefore, we have πΉπ·=12Γ4.6=2.3.cm In the same way, we can consider π·πΈ. By applying the triangle midsegment theorem again, we have that π·πΈ must be half the length of π΄π΅ (5.5 cm). Thus, we have π·πΈ=12Γ5.5=2.75.cm Finally, we can calculate the length of πΈπΉ in the same way. πΈπΉ must be half the length of π΄πΆ (6.2 cm). Thus, we have πΈπΉ=12Γ6.2=3.1.cm Hence, as πΉπ·=2.3cm, π·πΈ=2.75cm, and πΈπΉ=3.1cm, we can calculate the perimeter of β³πΈπΉπ· as perimeterofcmβ³πΈπΉπ·=πΉπ·+π·πΈ+πΈπΉ=2.3+2.75+3.1=8.15. We can give the answer that the perimeter of β³πΈπΉπ· is 8.15 cm. In the next example, we will see how we can apply our knowledge of the triangle midsegment theorem to help us prove geometrical properties within a given figure. In the given figure, πΈ and πΉ are the midpoints of π΄π΅ and π΄πΆ, respectively, π΅π·=12π΅πΆ, and π΅ lies on π·πΆ. What is the shape of πΈπΉπ΅π·? We are given that πΈ and πΉ are the midpoints of π΄π΅ and π΄πΆ respectively. Using the triangle midsegment theroem, we know that the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half its length. Hence, πΈπΉβ«½π΅πΆ and πΈπΉ=12π΅πΆ. We are asked to identify the shape of πΈπΉπ΅π·. πΈπΉπ΅π· appears to be a parallelogram; however, we must prove this to be the case. A parallelogram is defined as a quadrilateral with two pairs of opposite sides parallel. We have demonstrated by the triangle midsegment theorem that πΈπΉβ«½π΅πΆ, and since we are given that π΅ lies on π·πΆ, then πΈπΉβ«½π΅π·. We were given in the question that π΅π·=12π΅πΆ, and we proved that πΈπΉ=12π΅πΆ. Hence, πΈπΉ is congruent to π΅π·. We have now demonstrated that πΈπΉπ΅π· has a pair of opposite sides, πΈπΉ and π΅π·, that are parallel and congruent. Hence, πΈπΉπ΅π· must be a parallelogram. We will now see an example of how we can apply the converse of the triangle midsegment theorem to identify an unknown length. The perimeter of square π΄π΅πΆπ· is 352. Find π΄πΉ. AnswerIn this question, we are not given any information about the lengths of any line segments. However, we are given the information that the perimeter of this square is 352 length units. Given that the perimeter is the distance around the outside edge and that a square has 4 congruent sides, we can calculate the length of one side as lengthofonesideofthesquarelengthunits=352Γ·4=88. At this point, we might try to guess the length of π΄πΉ; however, in questions like this, we must apply our geometry knowledge to demonstrate and prove that our calculated length is correct. We can recall that the diagonals of a square bisect one another, so the diagonals π΄πΆ and π·π΅ are bisected at π. Therefore, we have that π is the midpoint of π΄πΆ. We can then apply the converse of the triangle midsegment theorem, which states that the line segment passing through the midpoint of one side of a triangle that is also parallel to another side of the triangle bisects the third side of the triangle. Hence, π΄π΅ is bisected by ππΉ, and π΄πΉ=πΉπ΅. We calculated earlier that the length of one side is 88 length units, so π΄π΅=88 length units. Since π΄πΉ=πΉπ΅, then πΉ is the midpoint of π΄π΅ and π΄πΉ=12π΄π΅. We can determine the length of π΄πΉ as π΄πΉ=12π΄π΅=12Γ88=44.lengthunits Therefore, we can give the answer that π΄πΉ is 44 length units. In the final example, we will see how we can use the converse of the triangle midsegment theorem to prove a geometric property in a given figure. In the given figure, which of the following is true?
AnswerIn the figure, we can observe that we have two pairs of congruent side lengths: π΄πΈ and πΈπΆ and π΅πΊ and πΊπΆ. Therefore, we can state that πΈ and πΊ are the midpoints of π΄πΆ and π΅πΆ respectively. We can recall that by the triangle midsegment theorem, the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half its length. Thus, in the figure, we have that πΈπΊβ«½π΄π΅β«½πΆπ·. We will now consider the choices that we are presented with and determine which of them is true. In choice A, we address the statement that πΈ is the midpoint of πΉπΊ. In fact, we have already proven that πΈ is the midpoint of π΄πΆ. We must be careful not to confuse the two line segments. Here, we cannot prove that πΈ is the midpoint of πΉπΊ. Next, letβs see the statement in choice π΅: πΉ is the midpoint of π΄π·. We can consider triangle π΄πΆπ·. Given that πΉπΈ passes though the midpoint of π΄πΆ and is parallel to πΆπ·, then by the converse of the triangle midsegment theorem, it must bisect the third side, π΄π·. Hence, π΄πΉ=πΉπ·, and πΉ is the midpoint of π΄π·. The statement given in choice B is true. In choice C, we need to determine if the statement that πΉπΊ=12π΄π΅ is true. Letβs consider β³π΄π΅πΆ. Applying the triangle midsegment theorem here would allow us to prove that πΈπΊ=12π΄π΅. Since πΉ can be observed not to lie on point πΈ and πΉβο«πΊπΈ, then πΉπΊ cannot also be the same length as πΈπΊ. Thus, this statement is not true. Finally, letβs consider choice D: πΆπ·=12π΄π΅. We cannot apply the triangle midsegment theorem or its converse to demonstrate that this statement is true. As with choice C, the line segment that can be demonstrated to be half the length of π΄π΅ is πΈπΊ, not πΆπ·. Therefore, the true statement is given in choice B: πΉ is the midpoint of π΄π·. We have seen how we can apply the triangle midsegment theorem and its converse. We can now summarize the key points.
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