Question

consider parallelogram g h j k below. note that g h j k has vertices g(4,4), h(-3,2), j(-5,-3), and k(2,-1). answer the following to determine if the parallelogram is a rectangle, rhombus, square, or none of these. (a) find the slope of \overline{g h} and the slope of a side adjacent to \overline{g h}. (b) find the length of \overline{g h} and the length of a side adjacent to \overline{g h}. give exact answers (not decimal approximations). (c) from parts (a) and (b), what can we conclude about parallelogram g h j k? check all that apply. \square g h j k is a rectangle. \square g h j k is a rhombus. \square g h j k is a square. \square g h j k is none of these.

Explanation:

Step1: Calculate the slope of \( \overline{GH} \)

The slope formula is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). For points \( G(4,4) \) and \( H(-3,2) \), we have \( x_1 = 4,y_1 = 4,x_2=-3,y_2 = 2 \).

$$ m_{GH}=\frac{2 - 4}{-3 - 4}=\frac{-2}{-7}=\frac{2}{7} $$

Step2: Calculate the slope of a side adjacent to \( \overline{GH} \)

Let's take \( \overline{HG} \) and \( \overline{HJ} \). For points \( H(-3,2) \) and \( J(-5,-3) \), using the slope formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \), with \( x_1=-3,y_1 = 2,x_2=-5,y_2=-3 \)

$$ m_{HJ}=\frac{-3 - 2}{-5+3}=\frac{-5}{-2}=\frac{5}{2} $$

Step3: Calculate the length of \( \overline{GH} \)

The distance formula is \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \). For \( G(4,4) \) and \( H(-3,2) \), we have \( x_1 = 4,y_1 = 4,x_2=-3,y_2 = 2 \)

$$ d_{GH}=\sqrt{(-3 - 4)^2+(2 - 4)^2}=\sqrt{(-7)^2+(-2)^2}=\sqrt{49 + 4}=\sqrt{53} $$

Step4: Calculate the length of a side adjacent to \( \overline{GH} \)

For \( H(-3,2) \) and \( J(-5,-3) \), using the distance formula \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \), with \( x_1=-3,y_1 = 2,x_2=-5,y_2=-3 \)

$$ d_{HJ}=\sqrt{(-5 + 3)^2+(-3 - 2)^2}=\sqrt{(-2)^2+(-5)^2}=\sqrt{4 + 25}=\sqrt{29} $$

Step5: Analyze the properties of the parallelogram

Since the product of the slopes \( m_{GH}\times m_{HJ}=\frac{2}{7}\times\frac{5}{2}=\frac{5}{7}
eq - 1 \), the adjacent sides are not perpendicular (so it's not a rectangle or square). Also, \( \sqrt{53}
eq\sqrt{29} \), so the adjacent sides are not equal (so it's not a rhombus).

Answer:

(a) Slope of \( \overline{GH}:\frac{2}{7} \); Slope of side adjacent to \( \overline{GH}:\frac{5}{2} \)
(b) Length of \( \overline{GH}:\sqrt{53} \); Length of side adjacent to \( \overline{GH}:\sqrt{29} \)
(c) \( GHJ K \) is none of these.