Objective:
To verify the mid-point theorem for a triangle.
Theorem :
The line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is congruent to one half of the third side.
Basic concepts and facts
1.Parallel Lines:
Two lines are parallel if they do not meet at any point.
2.Congruent Triangles:
Two triangles are congruent if their corresponding angles and corresponding sides are equal.
3.Similar triangles:
Two triangles are similar if their corresponding angles equal and their corresponding sides are in proportion.
Proof of theorem:
Given in the figure A :
AP=PB, AQ=QC.
To prove:
PQ || BC and PQ=1/2 BC
Plan:
To prove ▲ APQ ≅ ▲ QRC
Proof steps:
AQ=QC [midpoint]
∠ APQ = ∠QRC [Corresponding angles for parallel lines cut by an transversal].
∠PBR=∠QRC=∠APQ [Corresponding angles for parallel lines cut by an transversal].
∠RQC=∠PAQ [When 2 pairs of corresponding angles are congruent in a triangle, the third pair is also congruent.]
Therefore , ▲APQ ≅ ▲QRC
AP=QR=PB and PQ=BR=RC.
Since midpoints are unique, and the lines connecting points are unique, the proposition is proven.