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Question
Prove that the points A (1, -3), B (-3, 0) and C (4, 1) are the vertices of an isosceles right-angled triangle. Find the area of the triangle.
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Solution
AB =`sqrt((-3 -1)^2 + (0 +3)^2) = sqrt(16+9) = sqrt(25)` = 5
BC =`sqrt((4 + 3)^2 + (1 +0)^2)= sqrt(49+1)= sqrt(50) = 5sqrt(2)`
CA =`sqrt((1 -4)^2 + (-3 - 1)^2) = sqrt(9 + 16) = sqrt(25)` = 5
∵ AB = CA
A, B, C are the vertices of an isosceless triangle.
AB2 + CA2 = 25 + 25 = 50
BC2 = `(5sqrt(2))^2` = 50
∴ AB2 + CA2 = BC2
Hence, A, B, C are the vertices of a right-angled triangle.
Hence, ΔABC is an isosceles right-angled triangle.
Area of ΔABC = `(1)/(2) xx "AB" xx "CA"`
= `(1)/(2) xx 5 xx 5`
= 12.5 sq.units
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