Chords, Secants, and Tangents

When studying Geometry it is important to know the properties of the different shapes and figures that you may be studying. In this case we will discuss the properties of a circle. Let's begin with the definition of a circle. A circle is a curved line that creates a closed loop. This line is made up of infinitely many points and each of these points are an equal distance to the middle.

Properties

A circle has many parts and properties which are listed below.

• Center: The center of a circle is defined as the point in the middle of the circle. The points that make up the curve that is the circle are all equidistant from the center point.

• Radius: The radius is the distance from the center point to the circle, this distance is always equal in a circle. It is half the diameter. When working with the radius in geometry it is often denoted by $r$.

• Diameter: The diameter is the distance from one side of the circle to the other side of the circle. This distance is twice the radius. When working with the radius in geometry it is often denoted by $d$. It is important to remember that $d=2r$.

• Circumference: The circumference of a circle is the distance of the line that makes up the circle. Circumference is denoted by $C$ and can be calculated using the following formula: $C=2\pi r$.

• Area: The area of the space enclosed by the circle can be calculated with the following formula: $A_c=\pi r^2$.

• Chord: A chord is a line segment that links any two points on a circle.

• Tangent: A tangent line is a line that touches a circle at one point.

• Secant: A secant line is a line that goes through a circle and intersects the circle at two points.

Similar to a quadratic equation, a specific type of equation will create a circle when graphed. Unlike quadratic equations, the $x$ and $y$ terms are always squared. The equation of a circle is as follows: $$(x-h)^2+(y-k)^2=r^2,$$ where $h$ and $k$ are the $x$ and $y$ coordinates of the center point of the circle.

Chords, secants, and tangents are fundamental towards the understanding and identification of circles. Once you have a comprehensive understanding of these terms, you will be able to apply this knowledge in more advanced ways.

Let's fist consider a chord.

A cord is a segment whose endpoints lie on a circle.

In all the circles above, $\overline{AB}$ is a chord. In all the circles, $\overline{AB}$ is a segment whose endpoints lie on the circle.

If two chords intersect a circle, you are able to easily find the length of the segments. Consider the two chords that intersect the circle below.

When two chords intersect a circle, the product of the lengths of one of the chord is equal to the product of the lengths of the other chord.

In the circle above the formula looks like $\rightarrow$ $3\cdot{x}=5\cdot{4}$ $\rightarrow$ $3x=20$ $\rightarrow$ $x=\frac{20}{3}$

Next let's consider secants.

A secant is a line that contains a chord. Secants extend beyond the endpoints of a circle.

Intersecting Secant Theorem: When two secant segments are drawn to a circle from an external point, the product of one secant segment and its external segment equals the product of the other secant segment and its external segment.

$x\cdot{t}=y\cdot{s}$

Now let's consider tangents. A tangent is a line in the plane of a circle that intersects the circle in exactly one point, called the point of tangency.

In all the circles, $\overline{AC}$ are tangents, witch $B$ being the point of tangency.

Theorem: A line is tangent to a circle if and only if the line is perpendicular to the radius drawn to the point of tangency.

Because of this, the tangent will always form a right angle with its radius.

We know a triangle with a right angle classifies it as being a right triangle. Given the length of two of the sides, we can use the Pythagorean Theorem to check our answer.

$3^2+4^2=25$

$25=5^2$

$3^2+4^2=5^2$

We see the Pythagorean Theorem is true in this case, therefore it is correct.

Pictured below is a circle with the chord, secant, and tangent labeled.

Here's a great video reviewing the major properties of secants and chords:

Here's a somewhat unusual worked example of typical secant and tangent problems: