9.6. Summary

1. A circle is the collection of all points in a plane, which are equidistant from a fixed point in the plane.

A circle can be defined mathematically as a set of points that are all the same distance away from a particular point, known as the center. For example, if the center is point O and the distance from O to every point on the circle is r, then all points forming the circle will be r units away from O.

Think of a circle as the path traced by a point that always remains the same distance from a central point, like a dog walking around a fixed point in the yard on a longer leash.

2. Equal chords of a circle (or of congruent circles) subtend equal angles at the centre.

This statement means that if two chords in a circle are of equal length, the angles formed at the center of the circle by these chords will also be equal. This property helps in solving problems related to circles by relating chord lengths to angles.

Imagine two identical strings stretched between two points on a circular table. If you measure the angle each string makes with the center of the table, those angles will be the same since the strings (chords) are of equal length.

3. If the angles subtended by two chords of a circle (or of congruent circles) at the centre (corresponding centres) are equal, the chords are equal.

This property is the converse of the previous one. It states that if two angles at the center of the circle are equal, then the chords that subtend these angles must also be of equal length. This helps in deducing chord lengths based on angles.

Consider two identical pizza slices that have the same angle at the center of the pizza. The crusts of these slices (the chords) must also be of equal length, so the pieces of pizza are identical.

4. The perpendicular from the centre of a circle to a chord bisects the chord.

This means that if you drop a straight line (perpendicular) from the center of the circle to a chord, it will divide that chord into two equal halves. This concept is crucial for proving properties related to chords and their bisectors.

Imagine a tightrope walker who walks on a rope stretched between two poles. If a pole is positioned directly above the mid-point of the rope, it helps in keeping the rope balanced, similarly, a perpendicular from the center keeps the chord balanced.

5. The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.

This means that if you draw a line (that goes through the center) to cut a chord into two equal parts, that line must be at a right angle to the chord. This is another important property in circle geometry that helps in understanding the relationship between chords and their perpendicular bisectors.

Think of a person balancing on a seesaw. The point where the seesaw is balanced (the center) is like the center of the circle, and if you have two weights (the chord) at equal distance from the center, the seesaw will be perpendicular to the ground (the line drawn through the center).

6. Equal chords of a circle (or of congruent circles) are equidistant from the centre (or corresponding centres).

This property states that if two chords in a circle are of equal length, they will also be the same distance from the center of the circle. This ensures a consistent relationship between chord lengths and their distances from the center, aiding in geometric proofs.

If you think of two parallel shelves that are both the same height from the ground, the objects placed on these shelves (the chords) are equally distant from the ground (the center of the circle).

7. Chords equidistant from the centre (or corresponding centres) of a circle (or of congruent circles) are equal in length.

This describes how if two chords are at the same distance from the center of the circle, they must be equal in length. This helps in proving that certain pairs of chords are equal by measuring their distance from the center.

Imagine two rope lines attached to a central point, pulled taut at the same angle. Regardless of how long they are, if the distance from the center is the same, the lines (chords) will be equal.

8. If two arcs of a circle are congruent, then their corresponding chords are equal and conversely if two chords of a circle are equal, then their corresponding arcs (minor, major) are congruent.

This asserts that congruent arcs, meaning arcs that are identical in length and shape, have equal chords, and equally, if two chords are equal, the arcs they create will also be identical. This is fundamental in understanding the properties of circles and angular relations.

Picture two identical slices of a birthday cake; if the slices (chords) are the same size, the frosting-covered edges (arcs) will be identical. If the edges are the same, the two slices must also be equal.

9. Congruent arcs of a circle subtend equal angles at the centre.

This means that if two arcs are the same in size or shape, the angles they create at the center of the circle are also equal. It's an important principle used in circle theorems and helps establish relationships in geometric problems.

Think of two identical pizza slices. If you look at the vertex (center) of the pizza, the angle formed by the equal slices is the same, reflecting that both slices are congruent.

10. The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

This theorem highlights that the angle created at the center of the circle by an arc is always twice the angle created by that same arc when viewed from any other point on the circumference. This is particularly useful in solving complex geometric problems involving circles.

If you imagine a spotlight aimed at the center of a circular stage, the brighter light (angle at the center) seems more intense than the light viewed off-stage (at any other point), representing that the center view (angle) is always larger.

11. Angles in the same segment of a circle are equal.

This property states that if you take any two angles formed in the same segment of a circle, they will always be equal. This is foundational in proving and solving various circle theorems.

Envision two friends sitting in the same section of a movie theater and viewing scenes from the same angle. They both experience the same view (angle), representing how the angles in the segment of a circle function similarly.

12. Angle in a semicircle is a right angle.

This theorem specifies that any angle inscribed in a semicircle forms a right angle. This is a critical property when analyzing circles and measuring angles within them.

Imagine a rod placed straight through the center of a circular table. As you measure the angles created at either end of the rod across the circle, you’ll find they naturally form right angles, just like a perfect corner of a square.

13. If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment, the four points lie on a circle.

This property entails that if you can draw a line segment between two points that creates equal angles with any two other points on the same side, this configuration will form a circle containing all four points. This is a significant concept in both geometry and trigonometry.

Think of a square picnic table—if you and your friends are sitting at points on the opposite sides, the angles created at each corner between you and your friend directly across will always point towards the center of the table, which can represent a circle containing all four corners.

14. The sum of either pair of opposite angles of a cyclic quadrilateral is 180°.

This theorem states that if you have a quadrilateral where all four vertices lie on a circle, the sum of measures of each pair of opposite angles will equal 180 degrees. This highlights a key relationship in cyclic shapes.

Imagine a family portrait taken at a round table. No matter how you sit, if you measure the angles at the corners formed by the family members, adding up the angles directly across from each other will always result in a straight line (180°).

15. If the sum of a pair of opposite angles of a quadrilateral is 180°, the quadrilateral is cyclic.

This property is about the opposite of the previous theorem. It tells us that if we know a quadrilateral has opposite angles that add up to 180°, then we can conclude that all four points lie on a single circle, confirming it to be cyclic.

Consider a rectangular garden—since the corners (angles) add up perfectly to maintain a flat shape (180°), we can visualize that all corners lie on an imaginary circle encompassing the whole garden.

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