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Integrated math 1

Course: Integrated math 1 > Unit 18

Lesson 1: Distance and midpoints
Math
Integrated math 1
Analytic geometry
Distance and midpoints
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Getting ready for analytic geometry

Google Classroom
Analytic geometry relates geometric figures to the coordinate plane and algebraic representations. Let's review the coordinate plane, distance and displacement, slope, and a few helpful arithmetic skills to get ready.
Let’s refresh some concepts that will come in handy as you start the analytic geometry unit of the high school geometry course. You’ll see a summary of each concept, along with a sample item, links for more practice, and some info about why you will need the concept for the unit ahead.
There are a lot of sections in this article because analytic geometry pulls together a lot of ideas!
This article only includes concepts from earlier courses. There are also concepts within this high school geometry course that are important to understanding analytic geometry. If you have not yet mastered the Pythagorean theorem lesson, it may be helpful for you to review that before going farther into the unit ahead.

Points on a coordinate plane

What is this, and why do we need it?

We use a coordinate plane to show relative position in 2D space. We describe every point on the plane with an ordered pair in the form (x,y), where x represents the horizontal position, and y represents the vertical position. Points to the left of the
have negative x-coordinates, and points to the right have positive x-coordinates. Likewise, points below the origin have negative y-coordinates, and points above the origin have positive y-coordinates.

Practice

Problem 1
Use the following coordinate plane to write the ordered pair for each point.
A coordinate plane where point A is at negative three, negative one, point B is at three, negative one, and point C is at negative one, three.
PointOrdered pair
A(
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
,
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
)
B(
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
,
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
)
C(
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
,
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
)

For more practice, go to Points on the coordinate plane.

Where will we use this?

We will use points on a coordinate plane in almost every exercise in the analytic geometry unit! Here are a few of the exercises where reviewing the coordinate plane might be helpful:

Adding, subtracting, and squaring negative numbers

What is this, and why do we need it?

Negative numbers let us include direction information in a number. For example, a positive vertical change means we've gone up, but a negative vertical change means we've gone down. We'll be looking for distances and slopes between points on the coordinate plane. Points with negative coordinates are to the left of or below the
.

Practice

Problem 2.1
Add.
−7+4=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

For more practice, go to Adding negative numbers, Subtracting negative numbers, or Exponents with integer bases.

Where will we use this?

Here are a few of the exercises where reviewing negative numbers might be helpful:

Distance and displacement between points

What is this, and why do we need it?

Distance is how far apart two points are and is always non-negative. Displacement is the amount of change to go from one point to the other, including both distance and the direction of the change.
We often break distance and displacement into their horizontal and vertical parts. When we are only working with one direction of change (only horizontal or only vertical), then the distance is the absolute value of the displacement.
We use displacement to calculate slope, and we use the horizontal and vertical distances between points to find their total distance (with a little help from the Pythagorean theorem).

Practice

Problem 3.1
Complete the table of distances and displacements from point A to point B.
A coordinate plane where point A is at five, negative four and point B is at negative three, two.
DisplacementDistance
Horizontal
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
Vertical
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

For more practice, go to Distance between points: vertical and horizontal.

Where will we use this?

Here are a few of the exercises where reviewing distances and displacements might be helpful.

Simplifying square root expressions

What is this, and why do we need it?

For geometry, the square root function takes the area of a square as the input and give the length of a side of the square as an output. We'll use square root expressions when we use the Pythagorean theorem to find a distance. We'll use those distances to find area and perimeter of figures on the coordinate plane and to determine whether a point is part of a circle.

Practice

Problem 4.1
Simplify.
Remove all perfect squares from inside the square root.
180=

For more practice, go to Simplify square roots and Simplify square root expressions.

Where will we use this?

Here are a couple of the exercises where reviewing square root expressions might be helpful.

Scaling proportional relationships

What is this, and why do we need it?

Proportional relationships are two quantities where the ratio between the two quantities always stays the same.
Slope is a kind of proportional relationship that relates the vertical displacement (or change) to the horizontal displacement. We can scale the displacements between two points to find a third point between them that divides a line segment into lengths with a given ratio.

Practice

Problem 5
The double number line shows that to make 4 apple pies takes 7 kilograms (kg) of apples.
A number line labeled Pies that has tick marks from zero to four. Three evenly spaced tick marks are between zero and four.
A number line labeled Apples in kilograms has tick marks from zero to seven. Three evenly spaced tick marks are between zero and seven.
Select the double number line that correctly labels the number of kilograms of apples that are needed to make 1,2, and 3 pies.
Choose 1 answer:

For more practice, go to Create double number lines.

Where will we use this?

Here is an exercise where reviewing scaling proportional relationships might be helpful:

Slope

What is this, and why do we need it?

Slope is one way of measuring how steep a line is. We measure slope as ΔyΔx, which is the ratio of the vertical displacement to the horizontal displacement.
We can use the slope of a pair of lines to prove that they are parallel (or that they're not!). Then we can tell whether we can apply all of those relationships among the angles of figures with parallel lines. If use the slope to prove that two sides of a triangle are perpendicular, we can use trigonometric ratios to relate their angle measures and side lengths.

Practice

Problem 6.1
What is the slope of the line through (−4,2) and (3,−3)?
Choose 1 answer:

For more practice, go to Slope from two points, Slope-intercept intro, and Slope from equation.

Where will we use this?

Here are a few of the exercises where reviewing slope might be helpful.

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