Question

in given figure, PQR is an isosceles triangle with PQ = PR. S is a point on QR and T is a point on QP produced such that QT/PR=QR/QS,prove that
trianglePQS is similar to triangle TQR​

Answer 1

Answer:

In the given figure, ΔPQR is an isosceles triangle.

Therefore, side QP ≅ PR

It has been given that \frac{QT}{PR}=\frac{QR}{QS}PRQT=QSQR

Since QP = PR

Therefore, \frac{QT}{QP}=\frac{QR}{QS}QPQT=QSQR

By the theorem of similar triangles, If two triangles are similar then the corresponding sides of the similar triangles will be in the same ratio.

Therefore, ΔPQS ~ ΔTQR, in which corresponding sides are QT and QP, QR and QS.

Answer 2

Triangle PQS is similar to triangle TQR is proved.

Given:

PQ = PR

QT/PR=QR/QS

To prove:

ΔPQS≈ΔTQR

Proof:

• Triangle is a polygon having three sides and three vertices and angles.
• We can prove similarity of triangles using three criteria they are AA, SAS and SSS.
• If the corresponding angles of the two triangles are equal then they are said to be similar to each other under AA criteria.
• If the sides are proportional to the corresponding sides then they are said to be similar to each other under SSS criteria.
• If two sides are proportional to the corresponding sides and the angle between the sides are equal then they are said to be similar to each other under SAS criteria.

In ΔPQS and ΔTQR

QT/PR=QR/QS         (Given)

QT/PQ=QR/QS          (∵ PQ = PR)

∠Q=∠Q                   (Common angle)

Therefore, ΔPQS≈ΔTQR under SAS criteria.

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