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.


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

Four Triangles which are Congruent to One Another

To prove: ∆PMN ≅ LNM ≅ NQL ≅ MLR




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.


(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.

9th Grade Math

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