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, πΈ and πΉ are the midpoints of line segments π΄π΅ and π΄πΆ, respectively, π΅π· = (1/2) π΅πΆ, and π΅ lies on line segment π·πΆ. What is the shape of πΈπΉπ΅π·?

02:29

### Video Transcript

In the given figure, πΈ and πΉ are the midpoints of line segments π΄π΅ and π΄πΆ, respectively, π΅π· equals one-half π΅πΆ, and π΅ lies on line segment π·πΆ. What is the shape of πΈπΉπ΅π·?

We can begin this question by noting that the diagram shows the information that we have line segments that are divided into two congruent pieces, since πΈ is the midpoint of line segment π΄π΅ and πΉ is the midpoint of line segment π΄πΆ. The fact that we have these two midpoints might indicate that one of the triangle midsegment theorems can be applied: this one in particular that the line segment joining 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, line segment πΈπΉ is a line segment connecting the midpoints of two sides of a triangle. Therefore, it is parallel to the third side, which is π΅πΆ, and πΈπΉ must be half the length of this side π΅πΆ. And notice that we are given another side which is also equal to one-half π΅πΆ: this line segment, π΅π·.

So now, if we consider the quadrilateral πΈπΉπ΅π·, we know that this quadrilateral has a pair of congruent sides. And we can also say that these two sides are parallel, because we determined that line segment πΈπΉ was parallel to line segment π΅πΆ. And we know that π΅ lies on the line segment π·πΆ.

Now, the shape πΈπΉπ΅π· does look like a parallelogram, and in fact what we have shown here would prove this, because one way we can prove a quadrilateral is a parallelogram is by showing that one pair of opposite sides in a quadrilateral are both parallel and congruent. Therefore, we can give the answer that πΈπΉπ΅π· is a parallelogram.

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