Question

Prove ta line segment joining the midpoints of any two sides of a triangle is paralledl to thid side

Answer 1

To prove a segment joining the midpoints of two sides of a triangle is parallel to the third side, we use the midpoint theorem. By drawing triangles and showing similarity through SSS, we establish that alternate interior angles are equal, confirming the segment is parallel and half the length of the third side.

Step-by-step explanation:

To prove that a line segment joining the midpoints of any two sides of a triangle is parallel to the third side, let's consider a triangle ABC. Let D and E be the midpoints of sides AB and AC, respectively. According to the midpoint theorem, segment DE should be parallel to side BC and equal to half its length.

Here's the proof:

1. Draw triangle ABC with D and E as midpoints of sides AB and AC.
2. Since D and E are midpoints, AD = DB and AE = EC.
3. Triangles ADE and CBE are similar by the Side-Side-Side postulate, as AD = DB, AE = EC, and angle A is common in both triangles.
4. By corresponding angles of similar triangles, angle ADE is equal to angle BEC.
5. Therefore, line segment DE is parallel to BC because if alternate interior angles are equal, the lines are parallel according to Euclid's parallel postulate.
6. Futhermore, since the triangles are similar, the ratio of the sides is equal, which implies DE = (1/2)BC, proving the midpoint theorem.