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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