Show that the following points form an isosceles triangle.

(1, −2), (−5, 1) and (1, 4)

Formula used:

(1, −2), (−5, 1) and (1, 4)

Let the point be A (–5, 1) B (1, –2) and C (1, 4)

Distance of AB

⇒ AB = √ (1 – (–5))² + (–2 – 1)²)

⇒ AB = √ (1 + 5)² + (–2 – 1)²

⇒ AB = √ ((6)² + (–3)²)

⇒ AB = √ (36 + 9)

⇒ AB = √ 45 = 3√5

Distance of AC

⇒ AC = √ ((1 – (–5))² + (4 – 1)²)

⇒ AC = √ ((1 + 5)² + (4 – 1)²)

⇒ AC = √ ((6)² + (3)²)

⇒ AC = √ (36 + 9)

⇒ AC = √45 = 3√5

Distance of BC

⇒ BC = √ ((1 – 1)² + (4 – (–2)²)

⇒ BC = √ ((1 – 1)² + (4 + 2²)

⇒ BC = √ ((0)² + (6)²)

⇒ BC = √ (0 + 36)

⇒ BC = √ 36 = 6

We notice that AB = AC =3√5

∴ Points A, B and C are coordinates of an isosceles triangle.