question_answer

Show that \[\Delta \,ABC\], where \[A(-2,0),B(2,0),C(0,2)\] and \[\Delta \text{ }PQR\] where \[P(-4,0),Q(4,0),R(0,4)\] are similar triangles.

Answer:

Coordinates of vertices are

\[A(-2,0),B(2,0),C(0,2)\]

\[P(-4,0),Q(4,0),R(0,4)\]

\[AB=\sqrt{{{(2+2)}^{2}}+{{(0-0)}^{2}}}=4\,\,units\]

\[BC=\sqrt{{{(0-2)}^{2}}+{{(2-0)}^{2}}}\]

\[=\sqrt{4+4}=2\sqrt{2}\,units\]

\[CA=\sqrt{{{(-2-0)}^{2}}+{{(0-2)}^{2}}}\]

\[=\sqrt{8}=2\sqrt{2}\,\,units\]

\[PR=\sqrt{{{(0+4)}^{2}}+{{(4-0)}^{2}}}\]

\[=\sqrt{{{4}^{2}}+{{(4)}^{2}}}=4\sqrt{2}\,\,units\]

\[QR=\sqrt{{{(0-4)}^{2}}+{{(4-0)}^{2}}}\]

\[=\sqrt{{{4}^{2}}+{{(4)}^{2}}}=4\sqrt{2}\,\,units\]

\[PQ=\sqrt{{{(4+4)}^{2}}+{{(0-0)}^{2}}}\]

\[=\sqrt{{{(8)}^{2}}}=8\,\,units\]

We see that sides of \[\Delta \text{ }PQR\] are twice the sides of \[\Delta \text{ ABC}\].

Hence both triangles are similar.                            Hence Proved.