Answer:
So, the shortest distance from point H(3,2) to the line passing through J(-6,4) and K(-2,-4) is (16/√5) units.
Explanation:
To find the shortest distance from point H(3,2) to the line passing through points J(-6,4) and K(-2,-4), you can use the formula for the distance between a point and a line.
First, find the equation of the line passing through points J and K:
Calculate the slope (m) of the line:
m = (y2 - y1) / (x2 - x1)
m = (-4 - 4) / (-2 - (-6))
m = (-8) / (4)
m = -2
Use the point-slope form of a line equation with one of the points (let's use point J(-6,4)):
y - y1 = m(x - x1)
y - 4 = -2(x - (-6))
y - 4 = -2(x + 6)
Now, you have the equation of the line as:
y = -2x - 8
Next, you can find the perpendicular line passing through point H(3,2). The slope of this perpendicular line is the negative reciprocal of -2, which is 1/2.
Now, use the point-slope form with H(3,2):
y - 2 = (1/2)(x - 3)
Simplify:
y - 2 = (1/2)x - 3/2
Add 2 to both sides:
y = (1/2)x - 3/2 + 2
y = (1/2)x + 1/2
Now, you have the equation of the perpendicular line as:
y = (1/2)x + 1/2
The intersection point of these two lines will be the point on the line passing through J and K that is closest to H. So, you can set the two equations equal to each other and solve for x:
-2x - 8 = (1/2)x + 1/2
Now, solve for x:
-2x - (1/2)x = 1/2 + 8
(-4/2)x - (1/2)x = 17/2
(-5/2)x = 17/2
Now, divide by -5/2 (which is the same as multiplying by -2/5) to find x:
x = (17/2) * (-2/5)
x = -17/5
Now that you have x, you can find y using either of the equations. Let's use the equation of the perpendicular line:
y = (1/2)(-17/5) + 1/2
y = -17/10 + 1/2
y = -17/10 + 5/10
y = -12/10
y = -6/5
So, the point on the line passing through J and K that is closest to H is (-17/5, -6/5).
Now, find the distance between H(3,2) and this point:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Distance = √((-17/5 - 3)^2 + (-6/5 - 2)^2)
Distance = √((-17/5 - 15/5)^2 + (-6/5 - 10/5)^2)
Distance = √((-32/5)^2 + (-16/5)^2)
Distance = √((1024/25) + (256/25))
Distance = √(1280/25)
Distance = √(256/5)
Distance = (16/√5) units
So, the shortest distance from point H(3,2) to the line passing through J(-6,4) and K(-2,-4) is (16/√5) units.