Question

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directions: given the vertices, determine the quadrilaterals most specific classification: parallelogram, rectangle, rhombus, or square. justify your answer using the distance formula.

1. s(-9, 14), t(1, 10), u(-3, 0), v(-13, 4)

st:
stuv is a

Explanation:

Step1: Recall distance formula

The distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. We first calculate all side lengths.

Step2: Calculate length of $ST$

Substitute $S(-9,14)$ and $T(1,10)$:
$$ST=\sqrt{(1-(-9))^2+(10-14)^2}=\sqrt{(10)^2+(-4)^2}=\sqrt{100+16}=\sqrt{116}$$

Step3: Calculate length of $TU$

Substitute $T(1,10)$ and $U(-3,0)$:
$$TU=\sqrt{(-3-1)^2+(0-10)^2}=\sqrt{(-4)^2+(-10)^2}=\sqrt{16+100}=\sqrt{116}$$

Step4: Calculate length of $UV$

Substitute $U(-3,0)$ and $V(-13,4)$:
$$UV=\sqrt{(-13-(-3))^2+(4-0)^2}=\sqrt{(-10)^2+(4)^2}=\sqrt{100+16}=\sqrt{116}$$

Step5: Calculate length of $VS$

Substitute $V(-13,4)$ and $S(-9,14)$:
$$VS=\sqrt{(-9-(-13))^2+(14-4)^2}=\sqrt{(4)^2+(10)^2}=\sqrt{16+100}=\sqrt{116}$$

Step6: Calculate diagonal $SU$

Substitute $S(-9,14)$ and $U(-3,0)$:
$$SU=\sqrt{(-3-(-9))^2+(0-14)^2}=\sqrt{(6)^2+(-14)^2}=\sqrt{36+196}=\sqrt{232}$$

Step7: Calculate diagonal $TV$

Substitute $T(1,10)$ and $V(-13,4)$:
$$TV=\sqrt{(-13-1)^2+(4-10)^2}=\sqrt{(-14)^2+(-6)^2}=\sqrt{196+36}=\sqrt{232}$$

Step8: Analyze properties

All sides are equal ($\sqrt{116}$), so it is a rhombus. Diagonals are equal ($\sqrt{232}$), so it is also a rectangle. A quadrilateral that is both a rhombus and a rectangle is a square.

Answer:

Square