9. CIRCLES

You have already studied about circles and its parts in Class VI. Take a line segment PQ and a point R not on the line containing PQ. Join PR and QR. Then ∠ PRQ is called the angle subtended by the line segment PQ at the point R. What are angles POQ, PRQ and PSQ called in Fig. 9.2? ∠ POQ is the angle subtended by the chord PQ at the centre O, ∠ PRQ and ∠ PSQ are respectively the angles subtended by PQ at points R and S on the major and minor arcs PQ.

This chunk explains the concept of angles subtended by a chord. A chord is a line segment connecting two points on a circle. The angle formed at a point outside the chord (like point R) is termed as 'angle subtended by the chord'. Here, ∠ POQ, formed at the circle's center by the same endpoints as the chord, is mentioned as well as angles formed on the arcs of the circle by points R and S.

Think about a pizza. If you have a slice cut by a line (chord), and you're observing it from a point outside that slice, the angle you see formed at that outside point is similar to ∠ PRQ. The larger the slice of pizza you choose (the bigger the chord), the bigger the angle you see!

Let us examine the relationship between the size of the chord and the angle subtended by it at the centre. You may see by drawing different chords of a circle and angles subtended by them at the centre that the longer is the chord, the bigger will be the angle subtended by it at the centre. What will happen if you take two equal chords of a circle?

In this chunk, the relationship is established between the length of a chord and the angle it subtends at the center of the circle. As a chord increases in length, the angle at the center formed by the chord also increases. This observation leads to the idea that chords of equal length will subtend equal angles at the center.

Imagine a rubber band stretched into different lengths. When the band is short and stretched tight, the angle you can hold it at increases as you stretch it further. Just like in our example with the chords, when the rubber band is longer, you can see a wider stretch (larger angle) at the center.

Draw two or more equal chords of a circle and measure the angles subtended by them at the centre. You will find that the angles subtended by them at the centre are equal.

This part states that if two chords of equal length are drawn in a circle, the angles they subtend at the center of the circle will also be equal. This is a fundamental property of circles that can be proved mathematically through congruent triangles formed by the radii of the circle connecting to both chords.

Think of two identical ropes hanging from a tree. If you make them both the same length and attach them to the same point, when you pull them outwards to form angles, they will form the same angle at the tree (the analogy of the center of a circle!).

Theorem 9.1 : Equal chords of a circle subtend equal angles at the centre. Proof: You are given two equal chords AB and CD of a circle with centre O. You want to prove that ∠ AOB = ∠ COD. In triangles AOB and COD, OA = OC (Radii of a circle), OB = OD (Radii of a circle), AB = CD (Given). Therefore, ∆ AOB ≅ ∆ COD (SSS rule). This gives ∠ AOB = ∠ COD (Corresponding parts of congruent triangles).

This chunk provides the mathematical proof for Theorem 9.1 which states that equal chords subtend equal angles at the center of the circle. By using principles of congruence (SSS - side-side-side), it is shown that the two triangles formed by the equal chords and radii are congruent, thus establishing that their corresponding angles are equal.

Imagine two identical pizzas cut into two equal slices. When observing from the center, the angle those slices make (the angle subtended) will be exactly the same because the slices are equal in size, just like the equal chords in our theorem.

Theorem 9.2 : If the angles subtended by the chords of a circle at the centre are equal, then the chords are equal. Theorem 9.4 : The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.

Theorem 9.2 presents the converse of Theorem 9.1 — it states that if two angles subtended at the center are equal, then the corresponding chords are also equal. Theorem 9.4 introduces the concept that a line drawn from the center to bisect a chord will always meet the chord at a right angle, establishing another critical relationship.

Think of a seesaw at a playground. If you were to balance it perfectly, the lengths on both sides would be equal. This balance is similar to how chords function with angles; equal angles maintain equal chords, akin to making sure the seesaw is balanced from the center point.

Activity: Draw a circle on a tracing paper. Let O be its centre. Draw a chord AB. Fold the paper along a line through O so that a portion of the chord falls on the other. Let the crease cut AB at the point M. Then, ∠ OMA = ∠ OMB = 90° or OM is perpendicular to AB.

This chunk introduces an activity to visualize how a line drawn from the circle's center to a chord creates a right angle with the chord, meaning it is perpendicular. The concept of bisecting a chord is also introduced — the line from the center divides the chord into two equal lengths at the point where it intersects.

Think about cutting a cake. If you cut it perfectly through the middle from the top (center), the two halves will be equal. Here, the center of the cake is like our center O, and the cut represents the perpendicular line to the chord.

Let AB be a line and P be a point. The length of the perpendicular from a point to a line is the distance of the line from the point. A circle can have infinitely many chords. You may observe by drawing chords of a circle that longer chord is nearer to the centre than the smaller chord.

This chunk describes how the distance of chords from the center of the circle relates to their lengths. A longer chord is closer to the center, while shorter chords are farther away. The defined distance here is the shortest line (perpendicular) from a point (the center) to the line (the chord).

Imagine throwing a dart at a dartboard. The closer you get to the bullseye (center), the better your score (longer chord). As your scores (lengths) improve, you are getting closer to that central point!

A quadrilateral ABCD is called cyclic if all the four vertices of it lie on a circle. You will find a peculiar property in such quadrilaterals: the sum of either pair of opposite angles of a cyclic quadrilateral is 180º.

Here, a cyclic quadrilateral is defined as a four-sided figure where all corners touch the circumference of a circle. An important property of such shapes is that the sum of each pair of opposite angles equals 180 degrees. This property is foundational in geometry and has many practical applications.

Think of a round table with four friends sitting around it. If they sit in what can be called a cyclic manner and pair up, their sitting arrangements' angles can be thought of as 180° all around the table, making sure everyone has a clear view!

In this chapter, you have studied several points, including the relationship between chords, angles, and properties of circles.

This part encapsulates the key concepts discussed in the chapter, consisting of results regarding angles subtended by chords, equal chords, perpendicular lines from the center to chords, and properties of cyclic quadrilaterals.

These principles can be visualized in activities like drawing circles, juggling balls in patterns, or even constructing real-life circular features like pizza or wheels, solidifying how math relates to everyday life!

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