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.

$Four Triangles which are Congruent to One Another$

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.

9th Grade Math

From Four Triangles which are Congruent to One Another to HOME PAGE

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.

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.

[?]Subscribe To This Site

$XML RSS$

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.

E-mail Address

First Name

Then Don't worry — your e-mail address is totally secure. I promise to use it only to send you Math Only Math.