Respuesta :
Answer:
i) A = (3, 3), B = (12, 6), C = (6, 52) : Not orthogonal, ii) A = (-10, 5), B = (12, 16), C = (6, 52) : Not orthogonal, iii) A = (-8, 3), B = (12, 8), C = (18, 4) : Not orthogonal, iv) A = (12, -14), B = (-16, 21), C = (-11, 25) : Orthogonal, v) A = (-12, -19), B = (20, 45) : Impossible orthogonality, vi) A = (30, 20), B = (-20, -15) : Impossible orthogonality.
Step-by-step explanation:
The statement indicates that segments AB and BC must be orthogonal. Vectorially speaking, this can be expressed by using the following expression from Linear Algebra:
[tex]\overrightarrow {AB} \bullet \overrightarrow {BC} = 0[/tex]
[tex](AB_{x}, AB_{y})\bullet (BC_{x},BC_{y}) = 0[/tex]
[tex]AB_{x}\cdot BC_{x} + AB_{y}\cdot BC_{y} = 0[/tex]
Now, let is evaluate each choice:
i) A = (3, 3), B = (12, 6), C = (6, 52)
[tex]\overrightarrow {AB} = \vec B - \vec A[/tex]
[tex]\overrightarrow {AB} = (12, 6) - (3, 3)[/tex]
[tex]\overrightarrow {AB} = (12-3, 6-3)[/tex]
[tex]\overrightarrow {AB} = (9, 3)[/tex]
[tex]\overrightarrow {BC} = \vec C - \vec B[/tex]
[tex]\overrightarrow {BC} = (6, 52) - (12, 6)[/tex]
[tex]\overrightarrow {BC} = (6 - 12, 52 - 6)[/tex]
[tex]\overrightarrow {BC} = (-6, 46)[/tex]
[tex]\overrightarrow {AB} \bullet \overrightarrow {BC} = (9, 3)\bullet (-6, 46)[/tex]
[tex]\overrightarrow{AB} \bullet \overrightarrow {BC} = (9)\cdot (-6) + (3) \cdot (46)[/tex]
[tex]\overrightarrow{AB}\bullet \overrightarrow {BC} = 84[/tex]
AB and BC are not orthogonal.
ii) A = (-10, 5), B = (12, 16), C = (6, 52)
[tex]\overrightarrow {AB} = \vec B - \vec A[/tex]
[tex]\overrightarrow {AB} = (12, 16) - (-10, 5)[/tex]
[tex]\overrightarrow {AB} = (12+10, 16-5)[/tex]
[tex]\overrightarrow {AB} = (22, 11)[/tex]
[tex]\overrightarrow {BC} = \vec C - \vec B[/tex]
[tex]\overrightarrow {BC} = (6, 52) - (12, 16)[/tex]
[tex]\overrightarrow {BC} = (6 - 12, 52 - 16)[/tex]
[tex]\overrightarrow {BC} = (-6, 36)[/tex]
[tex]\overrightarrow {AB} \bullet \overrightarrow {BC} = (22, 11)\bullet (-6, 36)[/tex]
[tex]\overrightarrow{AB} \bullet \overrightarrow {BC} = (22)\cdot (-6) + (11) \cdot (36)[/tex]
[tex]\overrightarrow{AB}\bullet \overrightarrow {BC} = 264[/tex]
AB and BC are not orthogonal.
iii) A = (-8, 3), B = (12, 8), C = (18, 4)
[tex]\overrightarrow {AB} = \vec B - \vec A[/tex]
[tex]\overrightarrow {AB} = (12, 8) - (-8, 3)[/tex]
[tex]\overrightarrow {AB} = (12+8, 8-3)[/tex]
[tex]\overrightarrow {AB} = (20, 5)[/tex]
[tex]\overrightarrow {BC} = \vec C - \vec B[/tex]
[tex]\overrightarrow {BC} = (18, 4) - (12, 8)[/tex]
[tex]\overrightarrow {BC} = (18 - 12, 4 - 8)[/tex]
[tex]\overrightarrow {BC} = (6, -4)[/tex]
[tex]\overrightarrow {AB} \bullet \overrightarrow {BC} = (20, 5)\bullet (-6, -4)[/tex]
[tex]\overrightarrow{AB} \bullet \overrightarrow {BC} = (20)\cdot (-6) + (5) \cdot (-4)[/tex]
[tex]\overrightarrow{AB}\bullet \overrightarrow {BC} = -140[/tex]
AB and BC are not orthogonal.
iv) A = (12, -14), B = (-16, 21), C = (-11, 25)
[tex]\overrightarrow {AB} = \vec B - \vec A[/tex]
[tex]\overrightarrow {AB} = (-16,21) - (12, -14)[/tex]
[tex]\overrightarrow {AB} = (-16-12, 21+14)[/tex]
[tex]\overrightarrow {AB} = (-28, 35)[/tex]
[tex]\overrightarrow {BC} = \vec C - \vec B[/tex]
[tex]\overrightarrow {BC} = (-11,25) - (-16, 21)[/tex]
[tex]\overrightarrow {BC} = (-11+16, 25-21)[/tex]
[tex]\overrightarrow {BC} = (5, 4)[/tex]
[tex]\overrightarrow {AB} \bullet \overrightarrow {BC} = (-28,35)\bullet (5, 4)[/tex]
[tex]\overrightarrow{AB} \bullet \overrightarrow {BC} = (-28)\cdot (5) + (35) \cdot (4)[/tex]
[tex]\overrightarrow{AB}\bullet \overrightarrow {BC} = 0[/tex]
AB and BC are orthogonal.
v) A = (-12, -19), B = (20, 45)
It is not possible to determine the orthogonality of this solution, since point C is unknown.
vi) A = (30, 20), B = (-20, -15)
It is not possible to determine the orthogonality of this solution, since point C is unknown.
