Mathematics
Grade 10
15 min
Midsegments of triangles
Midsegments of triangles
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1
Introduction & Learning Objectives
Learning Objectives
Define and accurately identify the midsegment of a triangle.
State and apply the Triangle Midsegment Theorem to determine unknown side lengths.
Use the properties of midsegments to identify parallel lines and determine angle measures.
Solve algebraic problems involving the relationships between a midsegment and the third side of a triangle.
Use coordinate geometry (midpoint, slope, and distance formulas) to verify the Triangle Midsegment Theorem.
Construct a two-column proof using the properties of midsegments.
Ever seen the support structures on a bridge or a roof truss and wondered how engineers find the perfect center brace? 🏗️ The secret lies in the simple but powerful geometry of triangle midsegments!
In this tutorial, you will learn about a special...
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Key Concepts & Vocabulary
TermDefinitionExample
MidpointA point on a line segment that divides it into two equal, or congruent, segments.If M is the midpoint of segment AB, then AM = MB.
Midsegment of a TriangleA line segment that connects the midpoints of two sides of a triangle.In triangle ABC, if D is the midpoint of AB and E is the midpoint of BC, then segment DE is a midsegment.
Parallel LinesTwo lines in the same plane that are always the same distance apart and never intersect.The top and bottom edges of a ruler are parallel.
Third SideIn the context of a midsegment, this is the side of the triangle that is not connected to the midsegment's endpoints.If midsegment DE connects the midpoints of sides AB and BC, then side AC is the 'third side'.
Coordinate ProofA proof that uses figures in the c...
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Core Formulas
Triangle Midsegment Theorem (Parallelism)
A midsegment of a triangle is parallel to the third side of the triangle.
If DE is a midsegment connecting the midpoints of sides AB and BC in ΔABC, then DE is parallel to AC. This is written as DE || AC. This property is crucial for finding angle relationships, such as congruent corresponding angles.
Triangle Midsegment Theorem (Length)
A midsegment of a triangle is half as long as the third side of the triangle. Formula: Midsegment = (1/2) * (Third Side)
If DE is a midsegment and AC is the third side in ΔABC, their lengths are related by the formula DE = (1/2)AC, or equivalently, 2 * DE = AC. Use this to find an unknown length when the other is known.
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Challenging
A triangle is placed on a coordinate plane with vertices O(0, 0), P(2a, 2b), and Q(2c, 0). Let M be the midpoint of OP and N be the midpoint of PQ. What are the coordinates of M and N?
A.M(2a, 2b), N(2c, 0)
B.M(a, b), N(a+c, b)
C.M(a, b), N(c, 0)
D.M(a, b), N(a-c, b)
Challenging
In a coordinate proof of the Triangle Midsegment Theorem for ΔABC, after establishing the coordinates of the vertices and finding the coordinates of midpoints D and E, what is the most logical next step to prove the parallelism part of the theorem?
A.Calculate the slopes of segment DE and segment AC.
B.Calculate the lengths of segment DE and segment AC using the distance formula.
C.Use the Pythagorean theorem on ΔABC.
D.Find the coordinates of the third midpoint, F.
Challenging
In ΔABC, D is the midpoint of AB and E is the midpoint of AC. The length of DE is 2x + y and the length of BC is 7x - y. In the same triangle, F is the midpoint of BC, and the length of DF is x + y, while the length of AC is 40. Find the length of BC.
A.30
B.40
C.50
D.60
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