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Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of ΔPQR (see the given figure). Show that ΔABC ∼ ΔPQR. - CBSE Class 10 - Mathematics

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Question

Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of ΔPQR (see the given figure). Show that ΔABC ∼ ΔPQR.

Solution

Median divides the opposite side.

`:. BD = (BC)/2 and QM = (QR)/2`

Given that

`(AB)/(PQ) = (BC)/(QR)=(AD)/(PM)`

`=>(AB)/(PQ) = (1/2(BC))/(1/2QR) = (AD)/(PM)`

`=> (AB)/(PQ) = (BD)/(QM) = (AD)/(PM)`

In ΔABD and ΔPQM,

`(AB)/(PQ) = (BD)/(QM)= (AD)/(PM) ` (Proved above)

∴ ΔABD ∼ ΔPQM (By SSS similarity criterion)

⇒ ∠ABD = ∠PQM (Corresponding angles of similar triangles)

In ΔABC and ΔPQR,

∠ABD = ∠PQM (Proved above)

`(AB)/(PQ) = (BC)/(QR)`

∴ ΔABC ∼ ΔPQR (By SAS similarity criterion)

  Is there an error in this question or solution?

APPEARS IN

 NCERT Solution for Mathematics Textbook for Class 10 (2019 to Current)
Chapter 6: Triangles
Ex. 6.30 | Q: 12 | Page no. 141
Solution Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of ΔPQR (see the given figure). Show that ΔABC ∼ ΔPQR. Concept: Criteria for Similarity of Triangles.
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