Based on the Pythagorean theorem, a. Since [d(A, B)]²+ [d(A, C)]²= [d(B, C)]², it is a right triangle.
The Pythagorean theorem states that the square of the length of the longest side of a right triangle is equal to the sum of the squares of the lengths of the other two smaller sides of the right triangle.
Given the following:
A(3, -1),
B(5, -8),
C(-4, -3)
Using the distance formula, [tex]\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[/tex], find AB, BC, and AC.
AB = √[(5−3)² + (−8−(−1))²]
AB = √(4 + 49)
AB = √53 units
BC = √[(5−(−4))² + (−8−(−3))²]
BC = √(81 + 25)
BC = √106 units
AC = √[(3−(−4))² + (−1−(−3))²]
AC = √(49 + 4)
AC = √53 units
Applying the Pythagorean theorem, we have:
AB² + AC² = BC²
(√53)² + (√53)² = (√106)²
53 + 53 = 106
106 = 106
Thus, the triangle is a right triangle.
Therefore, we can conclude that, based on the Pythagorean theorem, a. Since [d(A, B)]²+ [d(A, C)]²= [d(B, C)]², it is a right triangle.
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