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Mathematics · Triangles · NCERT Examples

Example 8

A complete, independently verified solution for NCERT Example 8.

Mathematics · Chapter 6 · NCERT Examples

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NCERT Example

Example 8

similar triangles · medians · SAS similarity · SSS similarity

CM and RN are respectively the medians of △ABC and △PQR. If △ABC ∼ △PQR, prove that:

(i) △AMC ∼ △PNR

(ii) CM/RN = AB/PQ

(iii) △CMB ∼ △RNQ

Similar triangles ABC and PQR with medians CM and RN respectively.
Figure 6-33 from the supplied NCERT chapter.

Solution

Board-exam working:

Given △ABC ∼ △PQR, the correspondence is A ↔ P, B ↔ Q and C ↔ R.

Therefore, corresponding sides and angles are equal in ratio:

ABPQ=BCQR=CARP...(1)\frac{AB}{PQ}=\frac{BC}{QR}=\frac{CA}{RP}\qquad ...(1)
∠A=∠P,∠B=∠Q,∠C=∠R\angle A=\angle P,\quad \angle B=\angle Q,\quad \angle C=\angle R

(i) Since CM and RN are medians:

AB=2AM,PQ=2PNAB=2AM,\qquad PQ=2PN
AMPN=ABPQ=CARP\frac{AM}{PN}=\frac{AB}{PQ}=\frac{CA}{RP}

Also, ∠MAC = ∠NPR because ∠A = ∠P. Therefore, △AMC ∼ △PNR by SAS similarity.

(ii) From △AMC ∼ △PNR:

CMRN=CARP\frac{CM}{RN}=\frac{CA}{RP}

From (1):

CARP=ABPQ\frac{CA}{RP}=\frac{AB}{PQ}

Therefore:

CMRN=ABPQ\frac{CM}{RN}=\frac{AB}{PQ}

(iii) From (1) and part (ii):

CMRN=BCQR\frac{CM}{RN}=\frac{BC}{QR}

Also:

BM=AB2,QN=PQ2BM=\frac{AB}{2},\qquad QN=\frac{PQ}{2}
BMQN=ABPQ=CMRN\frac{BM}{QN}=\frac{AB}{PQ}=\frac{CM}{RN}

Thus:

CMRN=BCQR=BMQN\frac{CM}{RN}=\frac{BC}{QR}=\frac{BM}{QN}

Therefore, △CMB ∼ △RNQ by SSS similarity.

Final answer

(i) △AMC ∼ △PNR; (ii) CM/RN = AB/PQ; (iii) △CMB ∼ △RNQ.