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Prove that the points (5, -2), (-4, 3) ...

Prove that the points (5, -2), (-4, 3) and (10, 7) are the vertices of an isosceles right-angled triangle.

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Let the points are `A(5,-2), B(-4, 3) and C(10, 7)`.
`therefore" "AB^(2)=(-4-5)^(2)+(3+2)^(2)=(-9)^(2)+(5)^(2)= 81+ 25=106`
`" "BC^(2)=(10+4)^(2)+(7-3)^(2)=(14)^(2)+(4)^(2)=196+16=212`
`" "AC^(2)=(10-5)^(2)+(7+2)^(2) =(5)^(2)+(9)^(2)=25+81=106 `
Therefore, `" "AB=AC=sqrt(106)`
`and " "AB^(2)+AC^(2)=BC^(2)`
`thereforeDeltaABC` is an isosceles right-angled triangle.
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