Chapter 10 Mid-chapter Review What Is the Name of the Longest Chord in a Circle

Many objects that we run into in our lifestyle are 'round' in shape like a money, bangles, bottle caps, the Earth, wheels etc. In layman terms, the shape is ordinarily mentioned as a circumvolve. A airtight airplane figure, which is formed by the gear up of all those points which are equidistant from a fixed betoken in the same plane, is known equally a circumvolve.

The collection of all the points during a plane, which are at a hard and fast distance from a hard and fast point inside the airplane, is named a circle. In other words, a circle can exist described equally the locus of a bespeak moving in a plane, in such a way that its distance from a fixed point is always constant. A closed plane figure, which is formed past the fix of all those points which are equidistant from a stock-still point in the same plane, is known as a circle.

Circumference of Circle

The circumference of a circle is frequently defined as the distance around a circle. This tin exist understood with the aid of an example. Suppose a wire of length 5 m is aptitude in order that it forms a circumvolve. Hither, the circumference is equal to the length of the wire, i.e. 5 m. The length of the entire circumvolve is named its circumference.

Example: The radius of a round rug is four ft. What is the circumference?

Solution:

radius = 4 ft

Circumference Formula = 2πr

=> 2 * 3.14 * four = 25.12ft

Centre

A circumvolve may be a bend all of whose points dwell an equivalent plane and are at an equivalent distance from the centre. The fixed point is called the centre of the circle. The above figure shows the green colour dot called the centre.

Radius

The stock-still signal is named the heart of the circle and therefore the fixed distance is named the radius of the circumvolve. The constant distance betwixt whatsoever point on the circle and its middle is called the radius. Note that the line segment joining the heart and any bespeak on the circle is additionally called a radius of the circle.

Radius = Diameter/2 = d/2

Instance: The distance around a carousel is 21.98 yd. What is the radius?

Solution:

Circumference = 21.98 yd

Bore = 21.98 yd ÷ 3.14 = 7 yd

By Formula, Radius = Diameter/ii

Therefore,

radius = 3.5 yd

Diameter

A chord passing through the centre of the circumvolve is called the diameter of the circle. Two radii at 180 degrees angle are the diameter of the circle. Diameter is the largest chord of the circle and every bore has an equivalent length, which is adequate to twice the radius.

Diameter = two × radius

Case: If the bore of a circumvolve is 142.8 mm, then what is the radius?

Solution:

diameter = 142.viii mm

By Formula, diameter = two * r

radius = (142.8 ÷ 2)= 71.4 mm

Interior and Exterior of a Circle

A circle divides the plane on which information technology lies into three parts. They are:

• Within the circle, which is additionally chosen the inside of the circle. Point lying in the aeroplane of the circle such that its distance from its centre is less than the radius of the circle is known every bit the interior signal.
• On the circle and Points lying in the plane of the circle such that its distance from its centre is equal to the radius of a circle. In simple words, a set of points lying on the circle are points on the circumference of a circle.
• Outside the circle, which is additionally called the outside of the circle. Points lying in the plane of the circumvolve such that its distance from its centre is greater than the radius of the circle are exterior points.

Expanse of Track = π × (Outer Radius – Inner Radius)

Example: The inner circumference of a circular runway is 440 thousand, and the rails is 14 m wide. Calculate the
cost of levelling the runway at 25 paise/m2.

Solution:

Let the radius of the inner circumvolve exist r m.

Now,

Inner circumference = 440 m

⇒ 2πr = 440

⇒ 2 × 22/7 × r = 440

⇒ r = 440 × 744

⇒ Inner radius, r = 70

We know that the rails is 14 m broad.

∴ Outer radius (R) = (70 + 14) = 84 m

Area of the track = π(R− r)= π (842 – 702)

=> 22/7 × (7056 – 4900)

=> 6776 m2

Cost of levelling at 25 paise per square meter

=> 6676 × 25 = 169400 paise

or,

169400/100 = Rs.1694

Chord

A line segment joining two different points on the circumference of a circle is called a chord of the circumvolve. A circle can have any number of chords. Diameter is the largest chord of a circle. A chord of a circle may be a line joining two points of the circumference. A chord passes through the centre is named diameter. Note that Diameter is that the longest chord and every bore has an equivalent length, which is adequate to twice the radius.

Length of Chord = 2 × √(radius² − distance²)

Instance: Notice the length of the chord of a circle where the radius is 7 cm and perpendicular distance from
the chord to the heart is 4 cm?

Solution:

Given radius, r = 8 cm

and distance, d = three cm

Chord length = 2√(rⁱⁱ – d²)

⇒ Chord length = ii√(viii² – iii²)

⇒ Chord length = 2√(64 – 9)

⇒ Chord length = 2√55

⇒ Chord length = 2 × seven.416

or, chord length = 14.83 cm

Arcs

A piece of a circle betwixt two points is named an arc. A role of the circumference of the circle is known equally an arc. An arc is a continuous piece of the circumvolve. The arc upper AB arc is known equally the minor arc and the lower AB arc is the major arc. Now go through the circular region which is cut off from the rest of the circle by a secant or a chord. In a circumvolve, equal chords accept equal arcs.

Length of Arc = ii * π * r * angle / 360

Case: If the radius of a circle is v cm and the measure of the arc is 110˚, what is the length of the arc?

Solution:

Arc length = 2 * π * r * angle / 360

=> ii * 3.fourteen * 5 * 110/360°

=> 9.6 cm

Segment

The region betwixt a chord and either of its arcs is named segment of the circle. Role of a circle bounded by a chord and an arc is known equally a segment of the circle. The effigy given below depicts the major and pocket-size segments of the circumvolve. Here, the segment AQBA is with major arc AQB is called Major Segment while the segment APBA with modest arc APB is called minor segment.

Case: Discover the lengths of the arcs cutting off from a circumvolve of radius 12 cm by a chord 12 cm long. Also,
find the surface area of the minor segment.

Solution:

Let AB be the chord. Joining A and B to O, we become an equilateral triangle OAB.

Thus, nosotros have: ∠O = ∠A = ∠B = lx°

Length of the arc ACB: 2π × 12 × lx/360 = 4π = 12.56 cm

Length of the arc ADB:

Circumference of the circle – Length of the arc ACB

=> 2π × 12 – 4π = 20π cm = 62.80 cm

Now, Area of the modest segment:

Area of the sector – Area of the triangle

=> [π × (12)ii × 60/360 – 3√4 × (12)2] = thirteen.08 cm2

Sector

The region betwixt an arc and therefore the ii radii, joining the centre to the tiptop points of the arc is named a sector. A sector of a circle is the role bounded by ii radii and an arc of a circumvolve. In the beneath-given fig. AOB is a sector of a circle with O every bit the heart. The pocket-size arc corresponds to the small-scale sector and therefore the major arc corresponds to the main sector. When two arcs are equal, that is, each may exist a semicircle, then both segments and both sectors become equivalent and each is understood as a semicircular region.

Area of sector = θ/360 × πr^two

Case: The area of a sector with a radius of half dozen cm is 35.4 cm2. Calculate the angle of the sector.

Solution:

Area of sector = Angle * π * r * r/360°

Bending * π * 6 * half-dozen/360° = 35.iv

Angle = 35.four/36π × 360°

= 112.67°

Some Issues with Solution

Question ane. Every diameter of a circle is additionally a chord. Is that the converse of this statement also true?

Answer: Every diameter could also be a chord considering its endpoints lie on the circumference of the circle. It's the longest chord that passes through the centre of the circle.

Question ii. State difference betwixt sector and segment of a circle.

Answer: A sector is that the region between an arc and thus the ii radii, joining the centre to the highest points of the arc whereas a segment is that the region betwixt a chord of a circle and its associated arc.

Question 3. For a quadrilateral to go cyclic quadrilateral, sum of a pair of its reverse angles must be adequate to ________.

Answer: For a quadrilateral to become cyclic quadrilateral, sum of a pair of its opposite angles must be acceptable to 180°.

Question 4. What percentage circles are often drawn passing through iii non-collinear points?

Reply: I and just one circumvolve can be fatigued through three given not-collinear points.

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Source: https://www.geeksforgeeks.org/circles-and-its-related-terms-class-9-maths/