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Question
Prove that the points (0, 0), (5, 5) and (-5, 5) are the vertices of a right isosceles triangle.
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Solution
The distance d between two points `(x_1, y_1)` and `(x_2, y_2)` is given by the formula
`d = sqrt((x_1- x_2)^2 + (y_1 - y_2)^2)`
In an isosceles triangle there are two sides which are equal in length.
By Pythagoras Theorem in a right-angled triangle, the square of the longest side will be equal to the sum of squares of the other two sides.
Here the three points are A(0,0), B(5,5) and C(−5,5).
Let us check the length of the three sides of the triangle.
`AB = sqrt((0 - 5)^2 + (0 - 5)^2)`
`= sqrt((-5)^2 + (0 - 5)^2)`
`= sqrt(25 + 25)`
`AB = 5sqrt2`
`BC = sqrt((5 + 5)^2 + (5 - 5)^2)`
`= sqrt((10)^2 + (0)^2)`
`= sqrt(100)`
BC = 10
`AC = sqrt((0 + 5)^2 + (0 - 5)^2)`
`= sqrt((5)^2 + (-5)^2)`
`=sqrt(25 + 25)`
`AC = 5sqrt2`
Here, we see that two sides of the triangle are equal. So the triangle formed should be an isosceles triangle.
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