Prove that the points (1, –1) \(\big(\frac{-1}{2},\frac{1}{2}\big)\)

1.	Prove that the points (1, –1) \(\big(\frac{-1}{2},\frac{1}{2}\big)\) and (1, 2) are the vertices of an isosceles triangle.
Answer» Let P(1, -1), Q \(\big(\frac{-1}{2},\frac{1}{2}\big)\) and R(1,2) be the vertices of the ΔPQR. Then, PQ = \(\sqrt{\big(\frac{-1}{2}-1\big)^2+\big(\frac{1}{2}+1\big)^2}\) = \(\sqrt{\frac{9}{4}+\frac{9}{4}}\) = \(\sqrt{\frac{18}{4}}\) = \(\frac{3\sqrt2}{2}\) QR = \(\sqrt{\big(1+\frac{1}{2}\big)^2+\big(2-\frac{1}{2}\big)^2}\) = \(\sqrt{\frac{9}{4}+\frac{9}{4}}\) = \(\sqrt{\frac{18}{4}}\) = \(\frac{3\sqrt2}{2}\) PR = \(\sqrt{(1-1)^2+(2+1)^2}\) = \(\sqrt9\) = 3 ∵ PQ = QR, the triangle PQR is isosceles.

Prove that the points (1, –1) \(\big(\frac{-1}{2},\frac{1}{2}\big)\) and (1, 2) are the vertices of an isosceles triangle.

Answer»

Let P(1, -1), Q \(\big(\frac{-1}{2},\frac{1}{2}\big)\) and R(1,2) be the vertices of the ΔPQR.

Then, PQ = \(\sqrt{\big(\frac{-1}{2}-1\big)^2+\big(\frac{1}{2}+1\big)^2}\) = \(\sqrt{\frac{9}{4}+\frac{9}{4}}\) = \(\sqrt{\frac{18}{4}}\) = \(\frac{3\sqrt2}{2}\)

QR = \(\sqrt{\big(1+\frac{1}{2}\big)^2+\big(2-\frac{1}{2}\big)^2}\) = \(\sqrt{\frac{9}{4}+\frac{9}{4}}\) = \(\sqrt{\frac{18}{4}}\) = \(\frac{3\sqrt2}{2}\)

PR = \(\sqrt{(1-1)^2+(2+1)^2}\) = \(\sqrt9\) = 3

∵ PQ = QR, the triangle PQR is isosceles.

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