Consider parallelogram GHJK below.
H
G
K
Note that GHJK has vertices G(-3, 6), H(-7, -4), J(-2, -6), and K (2, 4).
Answer the following to determine if the parallelogram is a rectangle, rhombus, square, or none of these.
(a) Find the length of JK and the length of a side adjacent to JK.
Give exact answers (not decimal approximations).
Length of JK:
Length of side adjacent to JK: O
(b) Find the slope of JK and the slope of a side adjacent to JK.
Slope of JK:



Answer :

I can help you with that. Let's break down the information step by step: (a) Finding the length of JK and the length of a side adjacent to JK: 1. Length of JK: To find the length of JK, we can use the distance formula, which is: Distance = √[(x2 - x1)^2 + (y2 - y1)^2] Given the coordinates of points J(-2, -6) and K(2, 4), we can calculate the length of JK: JK = √[ (2 - (-2))^2 + (4 - (-6))^2 ] JK = √[ 4^2 + 10^2 ] JK = √[ 16 + 100 ] JK = √116 JK = 2√29 2. Length of a side adjacent to JK: Since JK is opposite to side GH, a side adjacent to JK would be either GH or HJ. Let's find the length of GH: Given the coordinates of points G(-3, 6) and H(-7, -4), we can calculate the length of GH: GH = √[ (-7 - (-3))^2 + (-4 - 6)^2 ] GH = √[ (-4)^2 + (-10)^2 ] GH = √[ 16 + 100 ] GH = √116 GH = 2√29 Therefore, the length of JK is 2√29 and the length of a side adjacent to JK (in this case GH or HJ) is also 2√29. (b) Finding the slope of JK and the slope of a side adjacent to JK: 1. Slope of JK: To find the slope of JK, we can use the formula: Slope = (y2 - y1) / (x2 - x1) Given the coordinates of points J(-2, -6) and K(2, 4), we can calculate the slope of JK: Slope of JK = (4 - (-6)) / (2 - (-2)) Slope of JK = 10 / 4 Slope of JK = 5/2 2. Slope of a side adjacent to JK: Since JK is opposite to side GH, a side adjacent to JK would be either GH or HJ. Let's find the slope of GH: Given the coordinates of points G(-3, 6) and H(-7, -4), we can calculate the slope of GH: Slope of GH = (-4 - 6) / (-7 - (-3)) Slope of GH = -10 / -4 Slope of GH = 5/2 Therefore, the slope of JK is 5/2 and the slope of a side adjacent to JK (in this case GH or HJ) is also 5/2. In conclusion, based on the calculations: - The length of JK is 2√29 and the length of a side adjacent to JK is 2√29. - The slope of JK is 5/2 and the slope of a side adjacent to JK is 5/2. Feel free to reach out if you need further clarification or assistance!