Question

state prove mid point theorem class9​

Answer 1

MidPoint Theorem Proof

MidPoint Theorem ProofIf the line segment adjoins midpoints of any of the sides of a triangle, then the line segment is said to be parallel to all the remaining sides, and it measures about half of the remaining sides. ... DE = (1/2 * BC).

Answer 2

Answer:

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.