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## High school geometry

### Course: High school geometry>Unit 7

Lesson 1: Graphs of circles intro

# Getting ready for conic sections

Practicing finding measurements in a circle, using the Pythagorean theorem, and completing the square will help us get ready for reasoning about conic sections (such as circles and parabolas).
Let’s refresh some concepts that will come in handy as you start the conic sections 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.
This article only includes concepts from earlier courses. There are also concepts within this high school geometry course that are important to understanding right triangles and trigonometry. If you have not yet mastered the Distance and midpoints lesson, it may be helpful for you to review that before going farther into the unit ahead.

### What is this, and why do we need it?

A circle is the collection of all points that are a certain distance from its center. We use the word radius both to mean that distance (a number) and to mean any segment (a geometric figure) with one endpoint on the center of the circle and one endpoint on its circumference.
Similarly, the diameter can mean either the widest distance across the circle, which is 2 times the radius, or any segment that passes through the center of the circle with both endpoints on the circumference of the circle.

### Practice

Problem 1
What is the radius and diameter of the following circle?
start text, c, m, end text
Diameter equals
start text, c, m, end text

For more practice, go to Radius and diameter.

### Where will we use this?

We use this vocabulary throughout the unit. Here is the first exercise where reviewing the radius and diameter might be helpful:

## Pythagorean theorem

### What is this, and why do we need it?

The Pythagorean theorem is a, squared, plus, b, squared, equals, c, squared, where a and b are lengths of the legs of a right triangle and c is the length of the hypotenuse. The theorem means that if we know the horizontal and vertical distances between any two points, we can find out the distance between the points. We can use the Pythagorean theorem to find the length of the radius of a circle, to derive the equation of a circle, and to derive the equation of a parabola.

### Practice

Problem 2.1
• Current
Find the value of n in the triangle shown below.
Right triangle where the short leg is four units, the long leg is six units. The hypotenuse is n units.

### Where will we use this?

Here are a few of the exercises where reviewing the Pythagorean theorem might be helpful:

## Completing the square

### What is this, and why do we need it?

There is a pattern when we square a binomial:
left parenthesis, x, plus, b, right parenthesis, squared, equals, x, squared, plus, 2, b, x, plus, b, squared
We complete the square when we have an equation like x, squared, plus, 2, b, x, equals, c, and we find the value b, squared to add to both sides. Then the left side of the equation becomes a perfect square.
Rewriting equations for circles by completing the square puts it back in the form of the Pythagorean theorem so that we can see the coordinates of the circle's center and the square of the radius.

### Practice

Problem 3.1
• Current
What is the missing constant term in the perfect square that starts with x, squared, minus, 20, x ?
A rectangle that is split into a square and twenty vertical shaded rectangle bars. The to side of the square portion of the rectangle is labeled x. The vertical bars are labeled negative twenty. The other side of the square and rectangle is x as well.

### Where will we use this?

Here are a few of the exercises where reviewing completing the square might be helpful: