A right angled isosceles triangle is a triangle with two sides of equal length and one right angle. To prove that the points (-3, 3), (3, -7) and (7,9) are the vertices of a right angled isosceles triangle, we can use the distance formula and the Pythagorean theorem.
The distance formula is used to calculate the distance between two points in a coordinate plane. It is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Using the distance formula, we can find that the distance between the points (-3, 3) and (3, -7) is:
d = √((3 - (-3))^2 + ((-7) - 3)^2) = √(6^2 + (-10)^2) = √(36 + 100) = √136
Similarly, we can find that the distance between the points (-3, 3) and (7, 9) is also √136 and the distance between the points (3, -7) and (7, 9) is √36.
Now, we can use the Pythagorean theorem, which states that in a right angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
The Pythagorean theorem can be written as:
c^2 = a^2 + b^2
Since d1 and d2 are equal, we can say that the triangle is isosceles.
So, using the distance formula and the Pythagorean theorem, we have proven that the points (-3, 3), (3, -7) and (7,9) are the vertices of a right angled isosceles triangle.