Prove that the points A(-3,0), B(1,-3) and C(4,1) are the vertices of an isosceles right triangle

Prove that the points A(-3,0), B(1,-3) and C(4,1) are the vertices of

1.	Prove that the points A(-3,0), B(1,-3) and C(4,1) are the vertices of an isosceles right triangle
Answer» Let A (-3, 0), B (1, -3) and C (4, 1) be the given points. Then,AB =\xa0{tex}\\sqrt { ( 1 - ( - 3 ) ) ^ { 2 } + ( - 3 - 0 ) ^ { 2 } } = \\sqrt { 4 ^ { 2 } + ( - 3 ) ^ { 2 } } = \\sqrt { 16 + 9 }{/tex}= 5BC =\xa0{tex}\\sqrt { ( 4 - 1 ) ^ { 2 } + ( 1 + 3 ) ^ { 2 } } = \\sqrt { 9 + 16 }{/tex}= 5 unitsand, CA =\xa0{tex}\\sqrt { ( 4 + 3 ) ^ { 2 } + ( 1 - 0 ) ^ { 2 } } = \\sqrt { 49 + 1 } = 5 \\sqrt { 2 }{/tex}unitsClearly, AB = BC. Therefore, {tex}\\triangle{/tex}ABC is isosceles.Also, AB2 + BC2 = 25 + 25 = (5{tex}\\sqrt2{/tex})2 = CA2{tex}\\therefore{/tex}{tex}\\triangle{/tex}ABC is right-angled at B.Thus, {tex}\\triangle{/tex}ABC is a right-angled isosceles triangle.