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## High school geometry

### 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 parenthesis, x, comma, y, right parenthesis, 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
start color #ca337c, A, end color #ca337cleft parenthesis
,
right parenthesis
start color #7854ab, B, end color #7854ableft parenthesis
,
right parenthesis
start color #0c7f99, C, end color #0c7f99left parenthesis
,
right parenthesis

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
• Current
minus, 7, plus, 4, equals

### 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
• Current
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
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
• Current
Simplify.
Remove all perfect squares from inside the square root.
root, start index, end index, equals

### 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 parenthesis, start text, k, g, end text, right parenthesis 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, comma, 2, comma 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 start fraction, delta, y, divided by, delta, x, end fraction, 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
• Current
What is the slope of the line through left parenthesis, minus, 4, comma, 2, right parenthesis and left parenthesis, 3, comma, minus, 3, right parenthesis?

### Where will we use this?

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

## Want to join the conversation?

• what is the mkdpoint (6,4),(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).
• this is very fat
• Any one else just utterly confused on all of this or just me
• anyone know what #3 is
• thank god i finally understand analytical geometry the day before my test
(1 vote)
• 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?
(1 vote)
• What is the slope of (-8,4) and (4,-6)
(1 vote)
• Slope = 𝛥𝑦∕𝛥𝑥 = [change in 𝑦]∕[change in 𝑥]

The change in 𝑦 is the difference between the 𝑦-coordinate of the second point and the 𝑦-coordinate of the first point:
𝛥𝑦 = −6 − 4 = −10

Similarly, the change in 𝑥 is the difference between the 𝑥-coordinate of the second point and the 𝑥-coordinate of the first point:
𝛥𝑥 = 4 − (−8) = 12

So, the slope of the line that goes through the points (−8, 4) and (4, −6) is
−10∕12 = −5∕6
(1 vote)