Answer:
Explanation:
To show that the points A(-1,-3), B(6, 1), and C(2,-5) form the vertices of a right angle triangle, we need to show that one of the angles of the triangle is a right angle, i.e., it measures 90 degrees.
We can use the Pythagorean theorem to check if the triangle is a right triangle. According to the Pythagorean theorem, the square of the length of the hypotenuse (the side opposite the right angle) of a right triangle is equal to the sum of the squares of the lengths of the other two sides.
First, let's calculate the lengths of the sides of the triangle:
AB = sqrt((6 - (-1))^2 + (1 - (-3))^2) = sqrt(49 + 16) = sqrt(65)
BC = sqrt((2 - 6)^2 + (-5 - 1)^2) = sqrt(16 + 36) = sqrt(52)
AC = sqrt((2 - (-1))^2 + (-5 - (-3))^2) = sqrt(9 + 4) = sqrt(13)
Next, let's check if the Pythagorean theorem holds true for this triangle:
If AB is the hypotenuse:
AB^2 = 65
BC^2 + AC^2 = 52 + 13 = 65
Since AB^2 = BC^2 + AC^2, we can conclude that angle BAC is a right angle.
Therefore, the points A(-1,-3), B(6, 1), and C(2,-5) form the vertices of a right angle triangle, with angle BAC measuring 90 degrees.