Getting ready for conic sections

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.

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$2 times the radius, or any segment that passes through the center of the circle with both endpoints on the circumference of the circle.

For more practice, go to Radius and diameter.

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

The Pythagorean theorem is $a^2+b^2=c^2$a, squared, plus, b, squared, equals, c, squared, where $a$a and $b$b are lengths of the legs of a right triangle and $c$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.

For more practice, go to Use Pythagorean theorem to find right triangle side lengths.

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

There is a pattern when we square a binomial:

We complete the square when we have an equation like $x^2+2bx=c$x, squared, plus, 2, b, x, equals, c, and we find the value $b^2$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.

For more practice, go to Completing the square (intro), Completing the square (intermediate), and Completing the square.

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