Four Triangles which are Congruent to One Another

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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

1. PN = (frac{1}{2})PQ.

2. LM = (frac{1}{2})PQ.

3. PN = LM.

4. Similarly, PM = NL.

5. In ∆PMN and ∆LNM,

(i) PN = LM

(ii) PM = NL

(iii) NM = NM.

6. Therefore, ∆PMN ≅ LNM.

7. Similarly, ∆NQL ≅ LNM.

8. Also, ∆MLR ≅ LNM.

9. Therefore, ∆PMN ≅ LNM ≅ NQL ≅ MLR. (Proved)

            Reason

1. N is the midpoint of PQ.

2. By the Midpoint Theorem.

3. From statement 1 and 2.

4. Proceeding as above.

5.

(i) From 3.

(ii) From 4.

(iv) Common side.

6. By SSS criterion of congruency.

7. Proceeding as above.

8. Proceeding as above.

9. From statements 6, 7 and 8.

9th Grade Math

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