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!
