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. |
From Four Triangles which are Congruent to One Another to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.