Respuesta :
Answer:
Since the length of the sides respects the Pythagorean theorem, these points form a right triangle.
Step-by-step explanation:
Distance between two points:
Suppose that we have two points, [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex]. The distance between them is given by:
[tex]D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
Right triangle:
Sum of the squares of the two smaller sides is equal to the square of the largest side(Pythagorean theorem).
Length of side PQ:
P (-2, -3), Q (4, 1)
[tex]a = \sqrt{(4 - (-2))^2+(1 - (-3))^2} = \sqrt{6^2 + 4^2} = \sqrt{52}[/tex]
Length of side PR:
P (-2, -3), R (2,4)
[tex]b = \sqrt{(2 - (-2))^2+(4 - (-3))^2} = \sqrt{4^2 + 7^2} = \sqrt{65}[/tex]
Length of side QR
Q (4, 1), R (2,4)
[tex]c = \sqrt{(2 - 4)^2+(4 - 1)^2} = \sqrt{2^2 + 3^2} = \sqrt{13}[/tex]
Pythagorean Theorem:
Smaller sides: a and c
Largest side: b
So
[tex]a^2 + c^2 = b^2[/tex]
[tex](\sqrt{52})^2 + (\sqrt{13})^2 = (\sqrt{65})^2[/tex]
[tex]52 + 13 = 65[/tex]
[tex]65 = 65[/tex]
Since the length of the sides respects the Pythagorean theorem, these points form a right triangle.
