Around we go with Geometry: Circles, Chords, and Arcs

# Geometry students in Aliso Viejo are currently learning the properties of circles.

In addition to the parts of the circle (radii, diameter, circumference, π, and area) learned earlier in the year, Geometry students are learning about chords, arcs, and secants and how to measure them.

## A Geometry tutor in Aliso Viejo will explain that a chord is a line segment where both ends of the segment are on the circle.

A secant is the line that extends from the chord. An arc is the degree of the circle between two points on the circumference of the circle.

The circle to the left shows two chords. One of these is a diameter to the circle (the one passing through the center.) The other chord is perpendicular to the diameter. When the diameter and chord are perpendicular, the diameter will bisect the chord. That is, the two pieces of the chord will be equal. Also, the two pairs of arcs will be equal. A common Geometry problem is shown above.

First, let’s solve for y, so we can find the length of the chord. Remember, in Geometry, the two chord segments are equal. Using algebra, we can set the two pieces equal to each other:

2y – 5 = y + 1

Add 5 to both sides to get:

2y = y + 6

Then subtract y from the right side to find the solution:

y = 6

Knowing that y equals 6, the total length of the chord is the two segments added together:

2y – 5 + y + 1

Substituting in 6 for y changes the equations to the below:

2(6) – 5 + 6 + 1

Using the order of operations, multiply first:

12 – 5 + 6 + 1

Then subtract and add as indicated to find that the total length of the chord is 14.

Now, let’s solve for x. We already know that the degree measure of the arc on the right is 52 degrees. Again, the diameter bisected the arc, so we can set the two arc segments equal to each other:

6x – 2 = 52

Using the rules of algebra, add two to both sides:

6x = 54

Then divide by 6:

x = 9

If you’re having a challenge remembering the difference between a chord and a secant, or can’t remember if a diameter is a chord, or if the radius or diameter is longer, you may want to have a Geometry tutor help you understand the properties of circles. This will certainly help you ace your next Geometry test!

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