Circle and Secant Segment Problems, 3

Category: Plane Geometry

"Published in Newark, California, USA"

PT is tangent to circle O. Secant BA is perpendicular to PT at P. If TA = 6 and PA = 3, find (a) AB, (b) the distance from O to AB, and (c) the radius of circle O.

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Solution:

Consider the given figure above

Photo by Math Principles in Everyday Life

Since PT is perpendicular to AP, then we can use Pythagorean Theorem in order to solve for the length of PT as follows

Next, let's analyze further the given figure as follows

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If you draw a line segment from point O which is perpendicular to AB, then AB will be bisected. Also, the perpendicular line is the distance of a chord or AB to the center of a circle.

If a theorem says "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.", then the working equation is
 

Substitute the values of the line segments in order to solve for the value of x as follows
 

 

 

 

 

 
Therefore, the length of a chord is
 

 

 

 
If OT is perpendicular to PT and and PT is perpendicular to PB, then OT is parallel to PB. In this case, the distance of AB to the center of a circle is
 

 

 
and the radius of a circle is
 

 

 

Circle and Secant Segment Problems, 2

Category: Plane Geometry

"Published in Newark, California, USA"

From the given figure, find the ratio x : y.

Photo by Math Principles in Everyday Life

Solution:

Consider the given figure above

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The given figure consists of two secant segments that are drawn in a circle from an external point. In the same figure, there are two chords that intersect inside a circle. In this case, we can solve for the values of x and y by using the two theorems. 

If the first theorem says "The measure of an angle formed by two chord that intersect inside a circle is equal to the half the sum of the measures of the intercepted arcs.", then the working equation is

If the second theorem says "The measure of an angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside a circle is equal to half the difference of the measures of the intercepted arcs.", then the working equation is

From the two working equations, if we add the two equations, y will be eliminated. The value of x is

---------------------

Substitute the value of x to either one of the working equations, we have

 

Therefore, the ratio of x and y is

 

Square, Rectangle, and Parallelogram Problems, 13

Category: Plane Geometry, Trigonometry

"Published in Newark, California, USA"

From the given figure, find ∠A + ∠B + ∠C. 

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Solution:

Consider the given figure above

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Since the given figure is a rectangle that consists of three squares, then we can solve for the values of ∠A, ∠B, and ∠C by using basic trigonometric functions. The given diagonals are the hypotenuses of a square and two rectangles, respectively. 

The value of ∠C is
 

 

 

 

The value of ∠B is
 

 

 

 

The value of ∠A is
 

 

 

 

 

Therefore,
 

 

Solving Trigonometric Equations, 10

Category: Trigonometry

"Published in Vacaville, California, USA"

Solve for the value of x for the equation:

Solution:

Consider the given equation above

Since there's a half-angle function in the given equation, then we need to convert it into single angle function first as follows

Take the inverse cosine on both sides of the equation, we have 

                                     or

Therefore, the values of x are

 
where n is the number of revolutions.