Question Video: Completing a Proof Using the Triangle Midsegment Theorem | Nagwa Question Video: Completing a Proof Using the Triangle Midsegment Theorem | Nagwa

# Question Video: Completing a Proof Using the Triangle Midsegment Theorem Mathematics • First Year of Preparatory School

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In the given figure, which of the following is true? [A] πΈ is the midpoint of line segment πΉπΊ. [B] πΉ is the midpoint of line segment π΄π·. [C] πΉπΊ = (1/2) π΄π΅ [D] πΆπ· = (1/2) π΄π΅

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### Video Transcript

In the given figure, which of the following is true? Option (A) πΈ is the midpoint of line segment πΉπΊ. Option (B) πΉ is the midpoint of line segment π΄π·. Option (C) πΉπΊ equals one-half π΄π΅. Or option (D) πΆπ· equals one-half π΄π΅.

Letβs begin by having a look at the figure. We could observe that there are two pairs of congruent line segments, with the first pair being the line segments π΄πΈ and πΈπΆ. The second pair of congruent line segments are π΅πΊ and πΊπΆ. Therefore, we can note that points πΈ and πΊ must be the midpoints of the line segments π΄πΆ and π΅πΆ, respectively. And because we have these midpoints, we can apply 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. So, if we consider triangle π΄π΅πΆ, the line segment πΈπΊ connecting the midpoints of the two sides must be parallel to the third side, π΄π΅. And we were already given that line segments π΄π΅ and πΆπ· were parallel, so we have three parallel line segments.

Now, letβs consider which of the four statements are true, taking option (A) first. πΈ is the midpoint of line segment πΉπΊ. Now, we have established that πΈ is a midpoint. But itβs the midpoint of line segment π΄πΆ. Line segment πΉπΊ is this line segment in the middle of the figure. And we donβt have any information to prove that πΈ is the midpoint of line segment πΉπΊ. Therefore, we canβt say that option (A) is true.

Next, we can consider the statement in option (B). πΉ is the midpoint of line segment π΄π·. We can find point πΉ and line segment π΄π· on the left side of the diagram. If we consider this line segment as a side of triangle π΄πΆπ·, we can determine something about this line segment. Recalling the theorem 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 can observe that the line segment πΈπΉ is a line segment passing through the midpoint of one side of the triangle and it is parallel to another side. Therefore, the third side, which is the line segment π΄π·, is bisected by line segment πΈπΉ. That means that point πΉ is the midpoint of line segment π΄π·. And so, the statement in option (B) is true.

However, itβs worth checking if either of options (C) or (D) is also a true statement. Option (C) says that πΉπΊ equals one-half π΄π΅. To see if this is true or not, we can look at triangle π΄π΅πΆ which is colored in green. By the first theorem here, since we have the midsegment πΈπΊ in this triangle, we know that πΈπΊ must be half the length of the parallel side π΄π΅. But the given statement doesnβt say that πΈπΊ is one-half π΄π΅; it says that πΉπΊ is. And we can see that the points πΉ and πΈ donβt lie on the same position. So, statement (C) is not true.

Finally, we can take the last statement that πΆπ· equals one-half π΄π΅. Line segment πΆπ· is here at the bottom of the figure. But we canβt apply any of the triangle midsegment theorems or any other theorems here to tell us that this is a true statement.

Therefore, the only statement that we can say is true is that in option (B): πΉ is the midpoint of line segment π΄π·.

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