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Circle & Its Secant

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Circle & Its Secant


Secant

From Latin: secare “to cut”
A line that intersects a curve or circle at two points

.

Try this Drag either orange dot. The blue line will always remain a secant to the circle, except that if the two points coincide, the secant becomes a tangent.

The blue line in the figure above is called the “secant to the circle c”.

As you move one of the points P,Q, the secant will change accordingly. If the two points coincide at the same point, the secant becomes a tangent, since it now touches the circle at just one point.

The line segment inside the circle between P and Q is called a chord.

Intersecting Secants

As shown in the figure on the right, when two secants intersect at a point outside the circle, there is an interesting relationship between the line segments thus formed.

Other definitions

In trigonometry, the secant of an angle in a right triangle is the ratio of the hypotenuse to the adjacent side. The reciprocal of cosine.

Intersecting Secants Theorem

When two secant lines intersect each other outside a circle, the products of their segments are equal.

(Note: Each segment is measured from the outside point)
Try this In the figure below, drag the orange dots around to reposition the secant lines. You can see from the calculations that the two products are always the same. (Note: Because the lengths are rounded to one decimal place for clarity, the calculations may come out slightly differently on your calculator.)

This theorem works like this: If you have a point outside a circle and draw two secant lines (PAB, PCD) from it, there is a relationship between the line segments formed. Refer to the figure above. If you multiply the length of PA by the length of PB, you will get the same result as when you do the same thing to the other secant line.

More formally: When two secant lines AB and CD intersect outside the circle at a point P, then

PA.PB = PC.PD

It is important to get the line segments right. The four segments we are talking about here all start at P, and some overlap each other along part of their length; PA overlaps PB, and PC overlaps PD.

Relationship to Tangent-Secant Theorem

In the figure above, drag point B around the top until it meets point A. The line is now a tangent to the circle, and PA=PB. Since PA=PB, then their product is equal to PA2. So:

PA2 = PC.PD

This is the Tangent-Secant Theorem.

Relationship to Tangent Theorems

If you move point B around until it overlaps A, the resulting tangent has a length equal to PA2. Similarly, if you drag D around the bottom to point C, the that tangent has a length of PC2. From the this theorem

PA2 = PC2

By taking the square root of each side:

PA = PC

confirming that the two tangents froma point to a circle are always equal.

Intersecting Secant Angles Theorem

 

The angle made by two secants intersecting outside a circle is half the difference between the intercepted arc measures.
Try this In the figure below, drag the orange dots to reposition the secants. Note how the angles are related. (Note: The angles are rounded off to whole numbers for clarity).

 

 

When two secants intersect outside a circle, there are three angle measures involved:

  1. The angle made where they intersect (angle APB above)
  2. The angle made by the intercepted arc CD
  3. The angle made by the intercepted arc AB

This theorem states that the angle APB is half the difference of the other two. Stated more formally:

∠P=(∠COD−∠AOB)/2

This is read as “The measure of the angle P is the measure of the arc CD minus the measure of the arc AB divided by 2”

Recall that the measure of an arc is the angle it makes at the center of the circle. To see this more clearly, click on “show central angles” in the diagram above.

It works for tangents too

The theorem still holds if one or both secants is a tangent. In the figure above, drag point C to the right until it meets A. The top line is now a tangent to the circle, and points A and C are in the same location. But the theorem still holds using the measures of the arcs CD and AB in the same way as before.

Make both lines into tangents in this way, and convince yourself the theorem still works.

 

Secant (sec) – Trigonometry function

In a right triangle, the secant of an angle is the length of the hypotenuse divided by the length of the adjacent side. In a formula, it is abbreviated to just ‘sec’.

 sec x=H/A

Of the six possible trigonometric functions, secant, cotangent, and cosecant, are rarely used. In fact, most calculators have no button for them, and software function libraries do not include them.

They can be easily replaced with derivations of the more common three: sin, cos and tan.
Secant can be derived as the reciprocal of cosine:

 sec x=1/( cos x)

The inverse secant function – arcsec

For every trigonometry function such as sec, there is an inverse function that works in reverse. These inverse functions have the same name but with ‘arc’ in front. So the inverse of sec is arcsec etc. When we see “arcsec A”, we interpret it as “the angle whose secant is A”.

sec 60 = 2.000 Means: The secant of 60 degrees is 2.000
arcsec 2.0 = 60 Means: The angle whose secant is 2.0 is 60 degrees.

Sometimes written as asec or sec-1

Large and negative angles

In a right triangle, the two variable angles are always less than 90° . But we can in fact find the secant of any angle, no matter how large, and also the secant of negative angles.

Graph of the secant function

Because the secant function is the reciprocal of the cosine function, it goes to infinity whenever the cosine function is zero.

The derivative of sec(x)

In calculus, the derivative of sec(x) is sec(x)tan(x). This means that at any value of x, the rate of change or slope of sec(x) is sec(x)tan(x).

 

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