Maths-

General

Easy

Question

# State and prove the Midsegment Theorem.

## The correct answer is: Hence proved

### SOLUTION:

HINT: State and prove the theorem.

Complete step by step solution:

Triangle midsegment theorem states that the line segment connecting the midpoints

of any 2 sides of a triangle is

- Is one half the length of the third side.
- Is parallel to the third side.

Proof:

In △ABC, we connect 2 midpoints D and E of 2 sides AB and BC.

Consider 2 triangles, △ABC and △DBE

We have,

AD = BD (since D is the midpoint)

CE = BE (since E is the midpoint)

and ∠ABC = ∠DBE by SAS similarity criterion.

⇒ △ ABC and △DBE are similar triangles.

We know that corresponding sides of similar triangles are proportional.

Here, we have proved the first part.

Now, Take 2 line segments DE and AC and a transversal BC cutting these 2 lines.

Since, △ABC and △DBE are similar triangles we have congruent corresponding

angles.

⇒ ∠BED corresponds to ∠BCA

⇒ ∠BED ⩭ ∠BCA

So by the converse of corresponding angles theorem, since a pair of corresponding

angles created by the transversal BC are congruent, we conclude that DE and AC

are parallel.

Hence proved.

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