Probability That A Randomly Selected Chord Exceeds The Length Of The Radius Of Circle

If a chord is selected at random on a fixed circle what is the probability that its length exceeds the radius of the circle?

  1. Assume that the distance of the chord from the center of the circle is uniformly distributed.
    A.   0.5 C.   0.866
    B.   0.667 D.   0.75
  2. Assume that the midpoint of the chord is evenly distributed over the circle.
    A.   0.5 C.   0.866
    B.   0.667 D.   0.75
  3. Assume that the end points of the chord are uniformly distributed over the circumference of the circle.
    A.   0.5 C.   0.866
    B.   0.667 D.   0.75


Smallest Part From The Circle That Was Divided Into Four Parts By Perpendicular Chords

Divide the circle of radius 13 cm into four parts by two perpendicular chords, both 5 cm from the center. What is the area of the smallest part.

01 - Circle tangent to a given line and center at another given line

Problem 1
A circle is tangent to the line 2x - y + 1 = 0 at the point (2, 5) and the center is on the line x + y = 9. Find the equation of the circle.

The Circle

Definition of circle
The locus of point that moves such that its distance from a fixed point called the center is constant. The constant distance is called the radius, r of the circle.

09 Dimensions of smaller equilateral triangle inside the circle

From the figure shown, ABC and DEF are equilateral triangles. Point E is the midpoint of AC and points D and F are on the circle circumscribing ABC. If AB is 12 cm, find DE.

Two equilateral triangles inside a circle


08 Circles with diameters equal to corresponding sides of the triangle

From the figure shown below, O1, O2, and O3 are centers of circles located at the midpoints of the sides of the triangle ABC. The sides of ABC are diameters of the respective circles. Prove that

$A_1 + A_2 + A_3 = A_4$


where A1, A2, A3, and A4 are areas in shaded regions.

Circles with centers at midpoints of sides of a right triangle


Conic Sections

Conic sections can be defined as the locus of point that moves so that the ratio of its distance from a fixed point called the focus to its distance from a fixed line called the directrix is constant. The constant ratio is called the eccentricity of the conic.

01 Arcs of quarter circles

Example 01
The figure shown below are circular arcs with center at each corner of the square and radius equal to the side of the square. It is desired to find the area enclosed by these arcs. Determine the area of the shaded region.

Intersection of circular quadrants


01 Rectangle of maximum perimeter inscribed in a circle

Problem 01
Find the shape of the rectangle of maximum perimeter inscribed in a circle.

03 Area Inside the Cardioid r = a(1 + cos θ) but Outside the Circle r = a

Example 3
Find the area inside the cardioid r = a(1 + cos θ) but outside the circle r = a.


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