The The midsegment of a triangle is a line which links the midpoints of two sides of the triangle. See Midsegment of a triangle. This page shows how to construct (draw) the midsegment of a given triangle with compass and straightedge or ruler. This construction uses Constructing the Perpendicular Bisector of a Line Segment to find the midpoints of the sides.
Printable step-by-step instructions
The above animation is available as a printable step-by-step instruction sheet, which can be used for making handouts or when a computer is not available.
Proof
The image below is the final drawing above.
| Argument | Reason | |
|---|---|---|
| 1 | EF is the perpendicular bisector of AB. | By construction. For proof see Constructing the perpendicular bisector of a line segment |
| 2 | S is the midpoint of AB | From (1). EF bisects AB. |
| 3 | DG is the perpendicular bisector of BC. | By construction. For proof see Constructing the perpendicular bisector of a line segment |
| 4 | T is the midpoint of BC | From (1). DG bisects BC. |
| 5 | ST is a midsegment of the triangle ABC. | From (2),(4). By definition. A midsegment of a triangle is a line linking the midpoint of two of its sides. See Midsegment of a triangle. |
- Q.E.D
Try it yourself
Click here for a printable worksheet containing two triangle midsegment problems. When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.Acknowledgements
Thanks to Aaron Strand of Carmel High School, Indiana for suggesting, reviewing, and proofreading this construction
Other constructions pages on this site
Lines
- Introduction to constructions
- Copy a line segment
- Sum of n line segments
- Difference of two line segments
- Perpendicular bisector of a line segment
- Perpendicular from a line at a point
- Perpendicular from a line through a point
- Perpendicular from endpoint of a ray
- Divide a segment into n equal parts
- Parallel line through a point (angle copy)
- Parallel line through a point (rhombus)
- Parallel line through a point (translation)
Angles
- Bisecting an angle
- Copy an angle
- Construct a 30° angle
- Construct a 45° angle
- Construct a 60° angle
- Construct a 90° angle (right angle)
- Sum of n angles
- Difference of two angles
- Supplementary angle
- Complementary angle
- Constructing 75° 105° 120° 135° 150° angles and more
Triangles
- Copy a triangle
- Isosceles triangle, given base and side
- Isosceles triangle, given base and altitude
- Isosceles triangle, given leg and apex angle
- Equilateral triangle
- 30-60-90 triangle, given the hypotenuse
- Triangle, given 3 sides (sss)
- Triangle, given one side and adjacent angles (asa)
- Triangle, given two angles and non-included side (aas)
- Triangle, given two sides and included angle (sas)
- Triangle medians
- Triangle midsegment
- Triangle altitude
- Triangle altitude (outside case)
Right triangles
- Right Triangle, given one leg and hypotenuse (HL)
- Right Triangle, given both legs (LL)
- Right Triangle, given hypotenuse and one angle (HA)
- Right Triangle, given one leg and one angle (LA)
Triangle Centers
Circles, Arcs and Ellipses
- Finding the center of a circle
- Circle given 3 points
- Tangent at a point on the circle
- Tangents through an external point
- Tangents to two circles (external)
- Tangents to two circles (internal)
- Incircle of a triangle
- Focus points of a given ellipse
- Circumcircle of a triangle
Polygons
- Square given one side
- Square inscribed in a circle
- Hexagon given one side
- Hexagon inscribed in a given circle
- Pentagon inscribed in a given circle
Non-Euclidean constructions
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