In triangle PQR, M is the midpoint of PQ and N is the midpoint of PR. If MN = 9 cm, what is QR?
A4.5 cm, because the midsegment is twice the third side
B9 cm, because the midsegment equals the third side
C18 cm, because the third side is twice the midsegment
DCannot be determined without knowing the angles
By the Midsegment Theorem, the segment connecting midpoints of two sides of a triangle is exactly half the length of the third side. If MN = 9, then QR = 2 × 9 = 18 cm. Option A reverses the relationship. Option B applies the theorem to say midsegment = third side (forgetting the factor of 1/2). Option D is wrong because the theorem gives an exact result regardless of angles — it depends only on the midpoint condition, which fixes the proportional relationship completely.
Question 2 Multiple Choice
A student confuses a midsegment with a median. What is the correct distinction between them?
AA midsegment goes from a vertex to the midpoint of the opposite side; a median connects midpoints of two sides
BA median goes from a vertex to the midpoint of the opposite side; a midsegment connects midpoints of two sides
CBoth terms describe the same segment — the difference is only in which textbook you use
DA midsegment connects midpoints of two sides and passes through the triangle's centroid
A median connects a vertex to the midpoint of the opposite side — it starts at a corner and ends at the midpoint of the far side. A midsegment connects the midpoints of two sides — both endpoints are midpoints, and no vertex is involved. These are completely different segments with different properties. A triangle has three medians (all meeting at the centroid) and three midsegments (forming the medial triangle). The Midsegment Theorem applies only to midsegments. A median is not half the length of any side in general.
Question 3 True / False
In any triangle, a midsegment connecting two side midpoints is always parallel to the third side.
TTrue
FFalse
Answer: True
Parallelism is one of the two guaranteed properties in the Midsegment Theorem, and it holds for any triangle regardless of shape (scalene, isosceles, right, obtuse). The coordinate proof makes this clear: when you compute the slopes of the midsegment and the third side, they are always equal. This parallelism is not a coincidence — it follows directly from the halving of coordinates at midpoints, which produces vectors in exactly the same direction as the full side.
Question 4 True / False
The midsegment of a triangle is equal in length to the third side, since both the midsegment and the third side span the same width of the triangle.
TTrue
FFalse
Answer: False
This is the most common error with the Midsegment Theorem. Although the midsegment is parallel to the third side (spanning the same 'direction'), it is exactly half the length — not equal. The midpoints of two sides lie at the halfway position along each side, so every horizontal and vertical component of the midsegment vector is exactly half of the corresponding component of the third side. The intuitive confusion arises because parallel segments that span the 'same' space can have different lengths — the midsegment is higher up in the triangle and therefore shorter.
Question 5 Short Answer
Using the coordinate proof approach, explain why the midsegment is exactly half the length of the third side.
Think about your answer, then reveal below.
Model answer: Because midpoints halve all coordinate differences. If the two endpoints of the third side differ by (Δx, Δy), then the midpoints of the two adjacent sides have x-coordinates and y-coordinates that each differ by half as much: (Δx/2, Δy/2). The length of a segment depends on its coordinate differences via the distance formula: √((Δx)² + (Δy)²). Halving each coordinate difference gives √((Δx/2)² + (Δy/2)²) = (1/2)√((Δx)² + (Δy)²), which is exactly half the length of the third side.
The factor of 1/2 is not a coincidence or something to memorize — it follows inevitably from the definition of midpoint (which halves each coordinate difference) combined with the distance formula (which is proportional to the coordinate differences). This is also why the medial triangle has exactly 1/4 the area of the original: each dimension is halved, and area scales as the square of linear dimensions, so (1/2)² = 1/4.