find the equation of the
line that describes all points equidistant from the points
(1, 2) and (4, 5).
the desired locus is the perpendicular bisector of the line segment.
So, its slope is -1 and it passes through (5/2,7/2)
y - 7/2 = -1(x - 5/2)
To find the equation of the line that describes all points equidistant from the points (1, 2) and (4, 5), we can follow these steps:
Step 1: Find the midpoint between the two given points.
Step 2: Calculate the slope of the line passing through the two given points.
Step 3: Find the negative reciprocal of the slope from step 2 to get the perpendicular slope.
Step 4: Use the midpoint and perpendicular slope to write the equation of the line in point-slope form.
Let's calculate it step-by-step:
Step 1: Find the midpoint between the two given points.
midpoint (xₘ, yₘ) = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
= ((1 + 4) / 2, (2 + 5) / 2)
= (5 / 2, 7 / 2)
= (2.5, 3.5)
So, the midpoint is (2.5, 3.5).
Step 2: Calculate the slope of the line passing through the two given points.
slope (m) = (y₂ - y₁) / (x₂ - x₁)
= (5 - 2) / (4 - 1)
= 3 / 3
= 1
So, the slope is 1.
Step 3: Find the negative reciprocal of the slope from step 2.
perpendicular slope = -1 / slope
= -1 / 1
= -1
So, the perpendicular slope is -1.
Step 4: Use the midpoint and perpendicular slope to write the equation of the line in point-slope form.
The equation of the line in point-slope form is:
y - yₘ = mₚ (x - xₘ)
where (xₘ, yₘ) is the midpoint and mₚ is the perpendicular slope.
Plugging in the values, we get:
y - 3.5 = -1 (x - 2.5)
Simplifying the equation, we get:
y - 3.5 = -x + 2.5
Rearranging the equation, we get:
y = -x + 6
Therefore, the equation of the line that describes all points equidistant from the points (1, 2) and (4, 5) is y = -x + 6.
To find the equation of a line that describes all points equidistant from two given points, you can use the concept of the perpendicular bisector of the line connecting these two points.
Here are the steps to find the equation of the line:
Step 1: Find the midpoint of the line segment connecting the two given points. The midpoint is the average of the x-coordinates and the average of the y-coordinates of the points.
Midpoint(x₁, y₁) = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
Given points: (1, 2) and (4, 5)
Midpoint = ((1 + 4) / 2, (2 + 5) / 2) = (2.5, 3.5)
Step 2: Determine the slope of the line connecting the two given points. The slope (m) of a line passing through two points (x₁, y₁) and (x₂, y₂) can be calculated using the formula:
m = (y₂ - y₁) / (x₂ - x₁)
Given points: (1, 2) and (4, 5)
m = (5 - 2) / (4 - 1) = 1
Step 3: The perpendicular bisector of the line connecting the two given points will have a negative reciprocal slope to the line connecting the points. So, the slope of the perpendicular bisector (m') can be calculated as:
m' = -1 / m
m' = -1 / 1 = -1
Step 4: Now, we need to find the equation of the line passing through the midpoint (2.5, 3.5) with the slope -1.
Using the point-slope form of the equation of a line:
y - y₁ = m' (x - x₁)
Substituting the values:
y - 3.5 = -1 (x - 2.5)
Simplifying the equation:
y - 3.5 = -x + 2.5
Rearranging the equation in the standard form:
x + y = 6
So, the equation of the line that describes all points equidistant from the points (1, 2) and (4, 5) is x + y = 6.