1-6
Midpoint and Distance
in the Coordinate Plane
Toolbox
1-6 (a)
1-6(b)
Holt McDougal Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Essential Questions
How do you develop and apply the formula
for midpoint?
How do you use the Distance Formula and
the Pythagorean Theorem to find the
distance between two points?
Holt McDougal Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
A coordinate plane is a plane that is
divided into four regions by a horizontal
line (x-axis) and a vertical line (y-axis) .
The location, or coordinates, of a point are
given by an ordered pair (x, y).
Holt McDougal Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
You can find the midpoint of a segment by
using the coordinates of its endpoints.
Calculate the average of the x-coordinates
and the average of the y-coordinates of the
endpoints.
Holt McDougal Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Holt McDougal Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Helpful Hint
To make it easier to picture the problem, plot
the segment’s endpoints on a coordinate
plane.
Holt McDougal Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Example 1: Finding the Coordinates of a Midpoint
Find the coordinates of the midpoint of PQ
with endpoints P(–8, 3) and Q(–2, 7).
= (–5, 5)
Holt McDougal Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Example 2: Finding the Coordinates of an Endpoint
M is the midpoint of XY. X has coordinates
(2, 7) and M has coordinates (6, 1). Find
the coordinates of Y.
Step 1 Let the coordinates of Y equal (x, y).
Step 2 Use the Midpoint Formula:
Holt McDougal Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Example 2 Continued
Step 3 Find the x-coordinate.
Set the coordinates equal.
Multiply both sides by 2.
12 = 2 + x
– 2 –2
Simplify.
10 = x
Simplify.
Subtract.
The coordinates of Y are (10, –5).
Holt McDougal Geometry
2=7+y
– 7 –7
–5 = y
1-6
Midpoint and Distance
in the Coordinate Plane
The Ruler Postulate can be used to find the distance
between two points on a number line. The Distance
Formula is used to calculate the distance between
two points in a coordinate plane.
Holt McDougal Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Example 3: Using the Distance Formula
Find FG and JK.
Then determine whether FG ≅ JK.
Step 1 Find the
coordinates of each point.
F(1, 2), G(5, 5), J(–4, 0),
K(–1, –3)
Holt McDougal Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Example 3 Continued
Step 2 Use the Distance Formula.
Holt McDougal Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
You can also use the Pythagorean Theorem to
find the distance between two points in a
coordinate plane. You will learn more about the
Pythagorean Theorem in Chapter 5.
In a right triangle, the two sides that form the
right angle are the legs. The side across from the
right angle that stretches from one leg to the
other is the hypotenuse. In the diagram, a and b
are the lengths of the shorter sides, or legs, of the
right triangle. The longest side is called the
hypotenuse and has length c.
Holt McDougal Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Holt McDougal Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Example 4: Finding Distances in the Coordinate Plane
Use the Distance Formula and the
Pythagorean Theorem to find the distance, to
the nearest tenth, from D(3, 4) to E(–2, –5).
Holt McDougal Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Example 4 Continued
Method 1
Use the Distance Formula. Substitute the
values for the coordinates of D and E into the
Distance Formula.
Holt McDougal Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Example 4 Continued
Method 2
Use the Pythagorean Theorem. Count the units for
sides a and b.
a = 5 and b = 9.
c2 = a2 + b2
= 52 + 92
= 25 + 81
= 106
c = 10.3
Holt McDougal Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Cornell Notes Summary
D-Describe what we did new in our notes?
(midpoint and distance formulas)
L- Describe what you learned. Use an action
verb and give an example
I- Describe what was interesting about this
mathematical concept. Is it somehow related
to another math concept?
Q- What questions do you have about
completing these problems?
Holt McDougal Geometry