Question

Prove that the points a(-3,0) ,b(1,-3) and c(4,1) are the vertices of an isosceles right angled triangle. find the area of this triangle.

Answer 1

Let A(-3,0) , B(1,-3) and C(4 , 1 ) be the given points.



A ( -3 , 0 ) and B ( 1 , -3 ).


Here,

X1 = -3 , Y1 = 0 and X2 = 1 , Y2 = -3.




Therefore,


AB = ✓(X2 - X1 )² + (Y2 - Y1 )²


AB = ✓ ( 1 + 3)² + ( -3 + 0 )²




AB = √ 16 + 9


AB = ✓25 = 5 units.



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B ( 1 , - 3 ) and C ( 4 , 1 )


Here,

X1 = 1 , Y1 = -3 and X2 = 4 , Y2 = 1.



Therefore,


BC = ✓(X2 - X1 )² + ( Y2 - Y1 )²



BC = ✓ ( 4 - 1 )² + ( 1 + 3 )²


BC = √25 = 5 units.



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A ( - 3 , 0 ) and C ( 4 , 1 )



Here,


X1 = -3 , Y1 = 0 and X2 = 4 , Y2 = 1.



Therefore,


AC = ✓( X2 - X1 )² + ( Y2 - Y1 )²



AC = ✓(4+3)² + (1 - 0)²



AC = ✓50 = 5√2 units.




Thus , AB = BC = 5 units.





Therefore, ∆ABC is isosceles .



Also, ( AB² + BC² ) = (5² + 5² ) = 50 units.


And,



AC² = (5√2)² = 50 units.


Thus , AB² + BC² = AC² .


This shows that ∆ABC is right angled triangle at B.



In triangle ABC , we have :


Base ( BC ) = 5 units and Height ( AB ) = 5 units


Therefore,


Area of triangle ABC = 1/2 × Base × Height



=> 1/2 × 5 × 5


=> 12.5 sq units.