Circles Chords And Tangents

Concept:
Circumference of a circle ( ) = , where is the radius.
When a square is inscribed in a circle, the diagonal of the square is equal to the diameter of the circle.
For a square with side , its diagonal .
Diameter of a circle ( ) = . Step 1: Find the radius of the circle Given circumference cm. We know . So, . Solve for : Step 2: Relate the diagonal of the inscribed square to the diameter of the circle When a square is inscribed in a circle, its vertices lie on the circle. The diagonal of this square is equal to the diameter of the circle. Diameter of the circle, . Substitute : So, the diagonal of the inscribed square, cm. Step 3: Find the side of the square Let the side of the inscribed square be . The diagonal of a square is related to its side by . We have . So, . Solve for : To rationalize the denominator (optional but good practice, and helps match options): multiply numerator and denominator by . The side of the square inscribed in the circle is cm.

Concept: This problem involves the relationship between the angle subtended by a chord at the center of a circle and the angles subtended by the same chord at points on the major and minor arcs. Step 1: Angle subtended by the chord at the center Let the circle have center O and radius . Let AB be a chord such that its length is equal to the radius, i.e., . Consider . We have (radius), (radius), and (given). Since all three sides are equal ( ), is an equilateral triangle. The angle subtended by the chord AB at the center O is . In an equilateral triangle, all angles are . So, . Step 2: Angle subtended at the major arc The angle subtended by an arc (or chord) at the center is double the angle subtended by it at any point on the remaining part of the circle (the major arc in this case). Let C be any point on the major arc. Then, . Step 3: Angle subtended at the minor arc Let D be any point on the minor arc. The points A, C, B, D in order form a cyclic quadrilateral ACBD. In a cyclic quadrilateral, the sum of opposite angles is . So, . We found . Therefore, . The angle is the angle subtended by the chord AB at a point on the minor arc. Thus, the angle subtended by this chord at the minor arc of the circle is .