Class 9 Exam > Class 9 Notes > Theory - To verify the mid-point theorem for a triangle, Math, Class 9
Theory - To verify the mid-point theorem for a triangle, Math, Class 9 PDF Download
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.
FAQs on Theory - To verify the mid-point theorem for a triangle, Math, Class 9
| 1. What is the mid-point theorem for a triangle? |  |
Ans. The mid-point theorem states that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.
| 2. How can we verify the mid-point theorem for a triangle? | |
Ans. To verify the mid-point theorem for a triangle, we need to draw a triangle, locate the midpoints of two sides, join the midpoints, and check if the line segment is parallel to the third side and half its length.
| 3. What is the significance of the mid-point theorem in triangle geometry? | |
Ans. The mid-point theorem is significant in triangle geometry as it helps in understanding and proving various properties of triangles. It provides insights into the relationship between the sides and midpoints of a triangle, facilitating further exploration of geometric concepts.
| 4. Can the mid-point theorem be applied to any type of triangle? | |
Ans. Yes, the mid-point theorem can be applied to any type of triangle, including equilateral, isosceles, and scalene triangles. It is a general property that holds true for all triangles.
| 5. Are there any real-life applications of the mid-point theorem? | |
Ans. Yes, the mid-point theorem finds practical applications in various fields such as construction, engineering, and architecture. It helps in determining the position of midpoints in structures, dividing line segments into equal parts, and designing symmetrical structures.
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