# Four Triangles which are Congruent to One Another

Here we will show that the three line segments which join the middle points of the sides of a triangle, divide it into four triangles which are congruent to one another.

Solution:

Given: In ∆PQR, L, M and N are the midpoints of QR, RP and PQ respectively.

To prove: ∆PMN ≅ LNM ≅ NQL ≅ MLR

Proof:

 Statement Reason 1. PN = $$\frac{1}{2}$$PQ. 1. N is the midpoint of PQ. 2. LM = $$\frac{1}{2}$$PQ. 2. By the Midpoint Theorem. 3. PN = LM. 3. From statement 1 and 2. 4. Similarly, PM = NL. 4. Proceeding as above. 5. In ∆PMN and ∆LNM,(i) PN = LM(ii) PM = NL(iii) NM = NM. 5.(i) From 3.(ii) From 4.(iv) Common side. 6. Therefore, ∆PMN ≅ LNM. 6. By SSS criterion of congruency. 7. Similarly, ∆NQL ≅ LNM. 7. Proceeding as above. 8. Also, ∆MLR ≅ LNM. 8. Proceeding as above. 9. Therefore, ∆PMN ≅ LNM ≅ NQL ≅ MLR. (Proved) 9. From statements 6, 7 and 8.

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