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### Course: High school geometry>Unit 6

Lesson 1: Distance and midpoints

# Getting ready for analytic geometry

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 $\left(x,y\right)$, 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.
PointOrdered pair
$A$$\left($
,
$\right)$
$B$$\left($
,
$\right)$
$C$$\left($
,
$\right)$

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
$-7+4=$

### 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 $B$.
DisplacementDistance
Horizontal
Vertical

### 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.
$\sqrt{\phantom{A}180}=$

### 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 $\left(\text{kg}\right)$ of apples.
Select the double number line that correctly labels the number of kilograms of apples that are needed to make $1,2,$ and $3$ pies.

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 $\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}$, 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 $\left(-4,2\right)$ and $\left(3,-3\right)$?

### Where will we use this?

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

## Want to join the conversation?

• Any one else just utterly confused on all of this or just me
• You're not alone.
• What is the midpoint of the line segment connecting (6,4) and (3,-8)?
• The midpoint of two points is the point that is halfway between them on the x-axis, and halfway between the two points on the y-axis as well. We can represent the midpoint of points (x1, y1) and (x2, y2) like this, where you take the average of each coordinate:
Mp = ((x1 + x2) / 2, (y1 + y2) / 2)
If you try it with the points (6, 4) and (3, -8), you should end up with (9/2, -2).
• what is in a burber
• toma's toes
• What grade would you typically learn this in?
• 8, 9, or 10 grade, depending on your math level. Maybe even less.
• I don't understand problem 4.3 - how did you get that result? I keep getting 70sqrt6z^5. Please anyone, explain!

And plus, I don't understand problem 5 - where did the 14 come from?? I was able to get the result correct beacuse I got decimals and not whole numbers. I got 1:4/7 ~~ 1:0.57 ; 2:1.14 ; 3:1.71
Because you know that 4/7 is approximately 0.57, and you get that result and multiply by 1,2,and 3 right?
• sqrt z^5 can be further simplified into z^2sqrt z
• Are we really gonna use all this in our day-to-day life?
• Yes. For example, finding the distance between 2 points on a world map.
• I just realized we are just math equations
math makes up particles. the higgs boson and the higgs equation gives particles mass. The particles interact, governed by math equations, moving as math tells them to do. The particles make up protons and neutrons, makes an atom which makes a molecule which makes cells which makes organ tissues which makes organs which make organ systems which makes humans. But, since the universe will last an infinite time, this means that in a long long forgotten and deserted world, out of pure chance some atoms might form into a brain in a jar with a supercomputer, by quantum tunneling which is basically math and now the brain can be simulated. It would look, hear, speak, taste, and feel everything as if it was on earth, and a normal human being but it would just be a naked brain, a last remnant of the universe, and technically speaking a human, a brain with a body is less unlikely that just a single naked brain and therefore you are just a brain in a jar, in a popup universe where nothing is real, a figment of your imagination. The last hope of the universe until your inevitable death by a rogue cylindar of pure depleted uranium hits the brain in the jar and it implodes into some pickles.
• how are 2 points, 9 units away from each other, equal the distance of 97 squared. i just came back from the practice. god i hate math
(1 vote)
• Hello everyone! I was wondering what the difference between the geometry FL BEST course and the other geometry courses is. I have to take a geometry course to graduate from highschool. Will the FL BEST course count? Thank you!
(1 vote)
• Hi Mary Miller! I think the main difference is that the FL BEST course is for building a foundation at an early age. An example would be taking the course virtually through platforms like this along with you regular school standards to get ahead and get honors. And other geometry courses typically follow the Common Core State Standards (CCSS) or other state-specific standards, which might have different focal points and instructional methods. If you are a high school student in Florida, taking the FL BEST Geometry course, then it will certainly count towards your graduation requirements because it is aligned with the state standards. If you are outside of Florida, the acceptance of this course might depend on your school's or state's specific graduation requirements.
The specific website I used for this information was
Florida Department of Education Website:
https://www.fldoe.org/