Find the distance from the point (3, 2) to the line y = 3x + 2.
Well, you can memorize a formula that Reiny knows or you can figure it out.
First sketch a graph
find the line perpendicular to the given line that passes through the given point.
the slope of the given line is 3
so the slope of our perpendicular line is -1/3
so
y = -x/3 + b is our perpendicular line but what is b?
put in (3,2)
2 = -3/3 + b
b = 3
so
y = -x/3 + 3
Now where do these lines cross?
y = -x/3 + 3
y = 3 x + 2
------------- subtract
0 = - 10/3 x + 1
10 x = 3
x = 3/10
then
y = 9/10 + 20/10 = 29/10
so
d^2 = (3-3/10)^2 +( 2 - 29/10)^2
Well, I hope you have your measuring tape ready, because we're about to find the distance! Now, let's start by finding a perpendicular line to y = 3x + 2 that passes through the point (3, 2).
The slope of the line y = 3x + 2 is 3. So, the slope of the perpendicular line will be the negative reciprocal of 3, which is -1/3.
Using the point-slope form of a line, we can write the equation of the perpendicular line:
y - 2 = (-1/3)(x - 3)
Now, let's rearrange this equation into slope-intercept form (y = mx + b):
y = (-1/3)x + 11/3
Now, we can find the intersection point between y = 3x + 2 and y = (-1/3)x + 11/3. Solving these two equations simultaneously, we find x = 1 and y = 5.
Therefore, the point of intersection is (1, 5).
Finally, we can calculate the distance between (3, 2) and (1, 5) using the distance formula:
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(1 - 3)^2 + (5 - 2)^2]
= √[(-2)^2 + 3^2]
= √[4 + 9]
= √13
The distance from the point (3, 2) to the line y = 3x + 2 is approximately √13 units.
To find the distance from a point to a line, we can use the formula for perpendicular distance.
Step 1: Find the slope of the given line.
In the equation y = 3x + 2, we can see that the coefficient of x is 3. Therefore, the slope of the line is 3.
Step 2: Find the equation of the line passing through the given point with a slope perpendicular to the given line.
Since the given line has a slope of 3, the perpendicular line will have a slope equal to the negative reciprocal of 3. So, the slope of the perpendicular line will be -1/3.
Using the point-slope form of a line, we can write the equation of the perpendicular line as:
y - 2 = (-1/3)(x - 3)
Simplifying, we get:
y - 2 = -1/3x + 1
y = -1/3x + 3
Step 3: Find the point of intersection between the given line and the perpendicular line.
We can solve the system of equations formed by the given line and the perpendicular line to find their point of intersection.
y = 3x + 2 ....(1)
y = -1/3x + 3 ....(2)
Substituting equation (2) into equation (1), we get:
-1/3x + 3 = 3x + 2
Multiplying both sides by 3 to remove the fraction:
-1x + 9 = 9x + 6
10x = -3
x = -3/10
Substituting this value back into equation (1):
y = 3(-3/10) + 2
y = -9/10 + 2
y = -9/10 + 20/10
y = 11/10
So, the point of intersection is (-3/10, 11/10).
Step 4: Calculate the distance between the given point (3, 2) and the point of intersection (-3/10, 11/10).
Using the distance formula between two points (x1, y1) and (x2, y2):
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Calculating the distance:
Distance = √((-3/10 - 3)^2 + (11/10 - 2)^2)
Distance = √((-33/10)^2 + (1/10)^2)
Distance = √((1089/100) + (1/100))
Distance = √(1090/100)
Distance = √10.9
Therefore, the distance from the point (3, 2) to the line y = 3x + 2 is approximately √10.9.
To find the distance from a point to a line, you can use the formula for the distance between a point and a line. First, we need to find the equation of a line perpendicular to the given line, passing through the given point (3, 2). Since perpendicular lines have slopes that are negative reciprocals of each other, the slope of the perpendicular line will be -1/3.
Using the point-slope form, we can write the equation of the perpendicular line as y - 2 = (-1/3)(x - 3). Simplifying this equation, we get y = (-1/3)x + 11/3.
Now, we have two lines - the given line y = 3x + 2 and the perpendicular line y = (-1/3)x + 11/3. Next, we find the intersection point of these two lines by setting their equations equal to each other:
3x + 2 = (-1/3)x + 11/3
Multiplying through by 3 to eliminate the fraction, we have:
9x + 6 = -x + 11
Combining like terms:
10x = 5
Dividing both sides by 10:
x = 1/2
Substituting this x-value back into one of the equations, we find the y-coordinate of the intersection point:
y = 3(1/2) + 2 = 5/2
So, the intersection point of the two lines is (1/2, 5/2).
To find the distance, we use the distance formula, which states that the distance between two points (x₁, y₁) and (x₂, y₂) is given by:
d = √((x₂ - x₁)² + (y₂ - y₁)²)
Substituting the coordinates of the intersection point (1/2, 5/2) and the given point (3, 2) into the formula:
d = √((1/2 - 3)² + (5/2 - 2)²)
Simplifying this expression:
d = √((-5/2)² + (1/2)²)
d = √(25/4 + 1/4)
d = √(26/4)
d = √(13/2)
Therefore, the distance from the point (3, 2) to the line y = 3x + 2 is √(13/2), or approximately 2.710.