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प्रश्न
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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उत्तर
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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Alia and Shagun are friends living on the same street in Patel Nagar. Shagun's house is at the intersection of one street with another street on which there is a library. They both study in the same school and that is not far from Shagun's house. Suppose the school is situated at the point O, i.e., the origin, Alia's house is at A. Shagun's house is at B and library is at C. |
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OR
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