Kerr
MPM2D
Analytic Geometry: Line Segments and Circles
2.2 Length of a Line Segment
The formula for the length of a line segment, which is derived from
the Pythagorean Theorem, is:
Examples:
1.
What kind of triangle is formed by the points A(3, 3), B(-1, 2) and
C(0, -2)? Express your answers in radical form.
AB = √(x2 – x1)2 + (y2 – y1)2
AB = √(3 – (-1))2 + (3 - 2)2
AB = √(4)2 + (1)2
AB = √16 + 1
AB = √17
Please show these important steps:
1. Write the formula.
2. Substitute variables into the formula.
3. Calculate the answer.
Kerr
MPM2D
Analytic Geometry: Line Segments and Circles
AC =
AC =
AC =
AC =
AC =
√(x2 – x1)2 + (y2 – y1)2
√(3 – 0)2 + (3 – (-2))2
√(3)2 + (5)2
√9 + 25
√34
BC = √(x2 – x1)2 + (y2 – y1)2
BC = √(-1 – 0)2 + (2 – (-2))2
BC = √(-1)2 + (4)2
BC = √1 + 16
BC = √17
Therefore, the triangle is isosceles.
2.
Calculate the distance between 5x - 2y = 10 and the point (2, 4.5).
Round your answer to one decimal place.
In order to this question you must
know that the distance between a
point and a line is the shortest
distance. Connect the point to the
line with a perpendicular line.
Step #1: Find the equation of the line that connects the point (2, 4.5)
to the line 5x - 2y = 10.
5x - 2y = 10
-2y = -5x + 10
y=
5
x-5
2
Kerr
MPM2D
Analytic Geometry: Line Segments and Circles
m=
m=
5
2
Since the lines are perpendicular, the slope of the line
connecting the point to the line is the negative reciprocal.
2
or -0.4
5
y = mx + b
4.5 = (-0.4)(2) + b
4.5 = -0.8 + b
5.3 = b
y = -0.4x + 5.3
5x - 2y = 10
y = -0.4x + 5.3
Substitute y = -0.4x + 5.3 into 5x - 2y = 10.
5x - 2(-0.4x + 5.3) = 10
5x + 0.8x - 10.6 = 10
5.8x = 20.6
x = 3.55
Substitute x = 3.55 into 5x - 2y = 10.
5x - 2y = 10
5(3.55) - 2y = 10
17.75 - 2y = 10
-2y = -7.75
y = 3.88
Once you have the
equation of the line, you
must find the point of
intersection of the two
lines using substitution or
elimination. This will give
you the second ordered
pair you need to calculate
the length.
Kerr
MPM2D
Analytic Geometry: Line Segments and Circles
Therefore the point of intersection is (3.55, 3.88).
= √(x2 – x1)2 + (y2 – y1)2
= √(3.55 – 2)2 + (3.88 – 4.5)2
= √(1.55)2 + (-0.62)2
= √2.4025 + 0.3844
= √2.7869
= 1.7
l
Now that we have two
ordered pairs, we can
substitute them into the
formula to solve for the
length between the point
and the line.
Hints for the homework:

Question #12 is exactly like example #2.
Homework: Page 86 #1c, 2d, 3, 4e, 5e, 6, 8, 11,
12ac, 13, 15