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] 𝐸 is the midpoint of line segment 𝐹𝐺. [B] 𝐹 is the midpoint of line segment 𝐴𝐷. [C] 𝐹𝐺 = (1/2) 𝐴𝐡 [D] 𝐢𝐷 = (1/2) 𝐴𝐡

04:56

Video Transcript

In the given figure, which of the following is true? Option (A) 𝐸 is the midpoint of line segment 𝐹𝐺. Option (B) 𝐹 is the midpoint of line segment 𝐴𝐷. Option (C) 𝐹𝐺 equals one-half 𝐴𝐡. Or option (D) 𝐢𝐷 equals one-half 𝐴𝐡.

Let’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 𝐴𝐸 and 𝐸𝐢. The second pair of congruent line segments are 𝐡𝐺 and 𝐺𝐢. Therefore, we can note that points 𝐸 and 𝐺 must be the midpoints of the line segments 𝐴𝐢 and 𝐡𝐢, 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 𝐴𝐡𝐢, the line segment 𝐸𝐺 connecting the midpoints of the two sides must be parallel to the third side, 𝐴𝐡. And we were already given that line segments 𝐴𝐡 and 𝐢𝐷 were parallel, so we have three parallel line segments.

Now, let’s consider which of the four statements are true, taking option (A) first. 𝐸 is the midpoint of line segment 𝐹𝐺. Now, we have established that 𝐸 is a midpoint. But it’s the midpoint of line segment 𝐴𝐢. Line segment 𝐹𝐺 is this line segment in the middle of the figure. And we don’t have any information to prove that 𝐸 is the midpoint of line segment 𝐹𝐺. Therefore, we can’t say that option (A) is true.

Next, we can consider the statement in option (B). 𝐹 is the midpoint of line segment 𝐴𝐷. We can find point 𝐹 and line segment 𝐴𝐷 on the left side of the diagram. If we consider this line segment as a side of triangle 𝐴𝐢𝐷, 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 𝐸𝐹 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 𝐴𝐷, is bisected by line segment 𝐸𝐹. That means that point 𝐹 is the midpoint of line segment 𝐴𝐷. And so, the statement in option (B) is true.

However, it’s worth checking if either of options (C) or (D) is also a true statement. Option (C) says that 𝐹𝐺 equals one-half 𝐴𝐡. To see if this is true or not, we can look at triangle 𝐴𝐡𝐢 which is colored in green. By the first theorem here, since we have the midsegment 𝐸𝐺 in this triangle, we know that 𝐸𝐺 must be half the length of the parallel side 𝐴𝐡. But the given statement doesn’t say that 𝐸𝐺 is one-half 𝐴𝐡; it says that 𝐹𝐺 is. And we can see that the points 𝐹 and 𝐸 don’t lie on the same position. So, statement (C) is not true.

Finally, we can take the last statement that 𝐢𝐷 equals one-half 𝐴𝐡. Line segment 𝐢𝐷 is here at the bottom of the figure. But we can’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): 𝐹 is the midpoint of line segment 𝐴𝐷.

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