Question

A quadrilateral has vertices A(−3, 1), B(1, 8), C(6, 7), and D(5, 2). What kind of quadrilateral is it? Explain using the vectors.

Answer 1

Answer on Question #55350 – Math – Analytic Geometry

A quadrilateral has vertices $A(-3, 1)$, $B(1, 8)$, $C(6, 7)$, and $D(5, 2)$. What kind of quadrilateral is it? Explain using the vectors.

Solution

Draw a rectangle ABCD in the Cartesian coordinate system

Consider the sides of the quadrilateral as vectors.

We find coordinates and moduli of vectors $\overrightarrow{AB}, \overrightarrow{BC}, \overrightarrow{CD}, \overrightarrow{DA}$.

Coordinates

$$\overrightarrow{AB} = (x_B - x_A; y_B - y_A) = (1 - (-3); 8 - 1) = (1 + 3; 8 - 1) = (4; 7);$$

modulus

$$\left| \overrightarrow{AB} \right| = \sqrt{x_{AB}^2 + y_{AB}^2} = \sqrt{4^2 + 7^2} = \sqrt{65}.$$

Coordinates

$$\overrightarrow{BC} = (x_C - x_B; y_C - y_B) = (6 - 1; 7 - 8) = (5; -1);$$

modulus

$$\left| \overrightarrow{BC} \right| = \sqrt{x_{BC}^2 + y_{BC}^2} = \sqrt{5^2 + (-1)^2} = \sqrt{26}.$$

Coordinates

$$\overrightarrow{CD} = (x_D - x_C; y_D - y_C) = (5 - 6; 2 - 7) = (-1; -5);$$

modulus

$$\left| \overrightarrow {C D} \right| = \sqrt {x _ {C D} ^ {2} + y _ {C D} ^ {2}} = \sqrt {(- 1) ^ {2} + (- 5) ^ {2}} = \sqrt {2 6}.$$

Coordinates

$$\overrightarrow {D A} = \left(x _ {A} - x _ {D}; y _ {A} - y _ {D}\right) = (- 3 - 5; 1 - 2) = (- 8; - 1);$$

modulus

$$\left| \overrightarrow {D A} \right| = \sqrt {x _ {D A} ^ {2} + y _ {D A} ^ {2}} = \sqrt {(- 8) ^ {2} + (- 1) ^ {2}} = \sqrt {6 5}.$$

As the vectors are not collinear, then the sides are not parallel. Notice that the sides of a quadrilateral are pairwise identical: AB=AD and CB=CD.

According to the definition, this quadrilateral is a deltoid, also known as a kite.

A kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other.

Answer: this quadrilateral is a deltoid.

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