Скачать или смотреть ICSE Class 9 Ch-11 Mid-Point Theorem Ex-11 Q. No. 10-12 From ML Aggarwal Part-3

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Ch-11 Mid-Point Theorem from ML Aggarwal's Class 9 ICSE
Textbook states that the line segment joining the midpoints of any two sides of a triangle is parallel to the third side and half its length. Conversely, a line drawn through the midpoint of one side of a triangle parallel to another side bisects the third side. This theorem is fundamental for solving geometry problems involving triangles and quadrilaterals.
Here are the key aspects of the Mid-Point Theorem:
Statement 1 (Direct Theorem): If you connect the midpoints of two sides of a triangle, the resulting line segment will be parallel to the third side and exactly half the length of the third side.
Example: In triangle ABC, if E and F are the midpoints of AB and AC respectively, then EF is parallel to BC and EF = ½ BC.
Statement 2 (Converse Theorem): If a line is drawn from the midpoint of one side of a triangle, and this line is parallel to another side, then this line will bisect the third side.
Example: If E is the midpoint of AB and the line EF is parallel to BC, then F must be the midpoint of AC.
Key applications and implications:
Solving Geometry Problems: The theorem helps in proving properties of other geometric figures, like showing that the figure formed by joining the midpoints of a square is also a square, or that the triangle formed by joining the midpoints of an isosceles triangle is also isosceles.
Congruent Triangles: It provides a foundation for understanding how joining midpoints can create four congruent triangles within a larger triangle.
Quadrilaterals: The theorem is extended to quadrilaterals, where it can be used to determine properties of lines connecting midpoints of their sides

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