Midpoint Theorem Converse (Motivate) MCQs Quiz | Class 9
This Class IX Mathematics (Code 041) quiz focuses on Unit IV: Geometry. Specifically designed to test your understanding of the Converse of the Midpoint Theorem, this exercise covers the concept that a line drawn through the midpoint of one side of a triangle parallel to another side bisects the third side. Complete the 10 questions below, click submit to check your score, and download the detailed PDF answer sheet for your revision records.
Understanding the Converse of Midpoint Theorem
In Class 9 Geometry, the Converse of the Midpoint Theorem is a fundamental concept used to prove properties of lines and triangles. While the Midpoint Theorem connects two midpoints to find a parallel line, the Converse starts with one midpoint and a parallel line to find the second midpoint.
The Statement: The line drawn through the midpoint of one side of a triangle, parallel to another side, bisects the third side.
Key Components of the Theorem
- Given Condition 1: A point is the midpoint of one side of a triangle.
- Given Condition 2: A line passes through this point and is parallel to a second side.
- Conclusion: The line will intersect the third side exactly at its midpoint (bisecting it).
Comparison: Midpoint Theorem vs. Converse
| Feature | Midpoint Theorem | Converse of Midpoint Theorem |
|---|---|---|
| Given | Midpoints of two sides | Midpoint of one side + Parallel line |
| Result | Line is parallel to third side and half its length | Line bisects the third side |
Practical Application Tips
When solving geometry problems, look for these clues to apply the Converse:
- Identify a triangle structure.
- Look for a known midpoint on one side (often marked with equality symbols like single or double ticks).
- Look for parallel arrow markings indicating a line is parallel to the base.
- If both exist, you can immediately conclude that the intersection on the other side is a midpoint.
Extra Practice Questions
Try solving these without options to test your understanding:
- 1. In triangle PQR, M is the midpoint of PQ. A line is drawn through M parallel to QR meeting PR at N. If PR = 12 cm, what is the length of PN?
- 2. In triangle ABC, D is the midpoint of AB. A line DE is parallel to BC and intersects AC at E. If AE = 2x + 3 and EC = 13, find x.
- 3. Can the Converse of Midpoint Theorem be applied to a right-angled triangle? (Yes/No)
- 4. In a trapezium ABCD where AB is parallel to DC, a line is drawn through the midpoint of AD parallel to AB. Does it bisect BC?
- 5. If a line divides two sides of a triangle in the same ratio, is it parallel to the third side? (Note: This relates to the Basic Proportionality Theorem, but is conceptually linked).
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