Circle and Secant Segment Problems

Category: Plane Geometry

"Published in Newark, California, USA"

Find the values of x and y in the given figure

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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. The other line with the same external point is tangent to 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 we can solve for the values of x and y which are the secant and external segments.

By using the line segments AO and CO, the value of x is

By using the line segments OA and OE, the value of y is

 

 

Circle and Inscribed Angle Problems

Category: Plane Geometry

"Published in Newark, California, USA"

Find the values of x and y in the given figure

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Solution:

The given figure consists of two inscribed angles. Inscribed angle is an angle of which the vertex is located along the arc of a circle. The sides of the inscribed angles are the chords of a circle. ∠PSR and ∠PTQ are inscribed angles of circle O.

Let's consider ∠PTQ and draw the line segments OP and OQ as follows
 

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If a theorem says "The measure of an inscribed angle is equal to half the measure of its intercepted arc.", then the value of x which is the measure of ∠PTQ is
 

 

 

Let's consider ∠PSR and draw the line segments OP and OR as follows

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If a theorem says "The measure of an inscribed angle is equal to half the measure of its intercepted arc.", then the value of y which is the measure of arc QR is

Concentric Circles Problems, 2

Category: Plane Geometry

"Published in Vacaville, California, USA"

Find the area of the circular section as shown in the figure

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Solution: 

If a small circle is tangent to the chord of a big circle, then the chord is bisected at the point of tangency. The radii of two circles are perpendicular to the point of tangency. To analyze more the problem, it is better to label further the figure as follows

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Let R₁ = be the radius of a big circle
      R₂ = be the radius of a small circle

If you connect the half of a chord and radii of two circles, then it becomes a right triangle. By Pythagorean Theorem, we can have the first working equation as follows

The area of a big circle is  .

The area of a small circle is .

The area of the circular section is
 

 

 

But

Therefore, the area of the circular section is

 

 

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