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The quadrangle with the vertices A(−3,5,6), B(1,−5,7), C(8,−3,−1), and D(4,7,−2) is a

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Square

Rectangle

Parallelogram

Trapezoid

answer is A.

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Detailed Solution

We will find the lengths of the four sides of the quadrangle using the distance formula.
seo

Substituting x1 = −3, y1 = 5, z1 = 6, x2 = 1, y2 = −5, and z2 = 7 in the distance formula

\sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2} + {(z_{2} - z)}^{2}}

, we get
⇒ AB =

\sqrt{{(1 - (- 3))}^{2} + {(- 5 - 5)}^{2} + {(7 - 6)}^{2}}

⇒ AB =

\sqrt{{(1 + 3)}^{2} + {(- 5 - 5)}^{2} + {(7 - 6)}^{2}}

⇒ AB =

\sqrt{4^{2} + {(- 10)}^{2} + 1^{2}}

⇒ AB =

\sqrt{16 + 100 + 1}

⇒ AB =

\sqrt{117}

Substituting x1 = 1, y1 = −5, z1 = 7, x2 = 8, y2 = −3, and z2 = −1 in the distance formula, we get
⇒ BC =

\sqrt{{(8 - 1)}^{2} + {[- 3 - (- 5)]}^{2} + {(- 1 - 7)}^{2}}

⇒ BC =

\sqrt{{(8 - 1)}^{2} + {[- 3 + 5]}^{2} + {(- 1 - 7)}^{2}}

⇒ BC =

\sqrt{7^{2} + 2^{2} + {(- 8)}^{2}}

⇒ BC =

\sqrt{49 + 4 + 64}

⇒ BC =

\sqrt{117}

Substituting x1 = 8, y1 = −3, z1 = -1, x2 = 4, y2 = 7, and z2 = −2 in the distance formula, we get
⇒ CD =

\sqrt{{(4 - 8)}^{2} + {[7 - (- 3)]}^{2} + {[- 2 - (- 1)]}^{2}}

⇒ CD =

\sqrt{{(4 - 8)}^{2} + {(7 + 3)}^{2} + {(- 2 + 1)}^{2}}

⇒ CD =

\sqrt{{(- 4)}^{2} + 10^{2} + {(- 1)}^{2}}

⇒ CD =

\sqrt{16 + 100 + 1}

⇒ CD =

\sqrt{117}

Substituting x1 = 4, y1 = 7, z1 = -2, x2 = -3, y2 = 5, and z2 = 6 in the distance formula, we get
⇒ DA =

\sqrt{{(- 3 - 4)}^{2} + {(5 - 7)}^{2} + {[6 - (- 2)]}^{2}}

⇒ DA =

\sqrt{{(- 3 - 4)}^{2} + {(5 - 7)}^{2} + {(6 + 2)}^{2}}

⇒ DA =

\sqrt{{(- 7)}^{2} + {(- 2)}^{2} + 8^{2}}

⇒ DA =

\sqrt{49 + 4 + 64}

⇒ DA =

\sqrt{117}

We can observe that all the four sides are equal to

\sqrt{117}

.
Now, we will find the lengths of the diagonals of the quadrangle.
Substituting x1 = 8, y1 = −3, z1 = -1, x2 = -3, y2 = 5, and z2 = 6 in the distance formula, we get
⇒ AC =

\sqrt{{(- 3 - 8)}^{2} + {[5 - (- 3)]}^{2} + {[6 - (- 1)]}^{2}}

⇒ AC =

\sqrt{{(- 3 - 8)}^{2} + {(5 + 3)}^{2} + {(6 + 1)}^{2}}

⇒ AC =

\sqrt{{(- 11)}^{2} + 8^{2} + 7^{2}}

⇒ AC =

\sqrt{121 + 64 + 49}

⇒ AC =

\sqrt{234}

Substituting x1 = 4, y1 = 7, z1 = -2, x2 = 1, y2 = -5, and z2 = 7 in the distance formula, we get
⇒ BD =

\sqrt{{(1 - 4)}^{2} + {(- 5 - 7)}^{2} + {[7 - (- 2)]}^{2}}

⇒ BD =

\sqrt{{(1 - 4)}^{2} + {(- 5 - 7)}^{2} + {(7 + 2)}^{2}}

⇒ BD =

\sqrt{{(- 3)}^{2} + {(- 12)}^{2} + 9^{2}}

⇒ BD =

\sqrt{9 + 144 + 81}

⇒ BD =

\sqrt{234}

Therefore, we can observe that the diagonals are equal to

\sqrt{234}

.
Since the four sides of the quadrangle are equal to each other, and the diagonals are also equal to each other, the quadrangle with the vertices A(−3,5,6), B(1,−5,7), C(8,−3,−1), and D(4,7,−2) is a square.
Thus, the correct option is option (1).

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