Chapter: – GPS Geometry

TOPIC: – Properties of shapes

What is GPS Geometry?

The mathematical model of GPS observation consists of various stochastic and functional components. It includes various components which have properties of polygons, right triangles, coordinate geometry, circles, right triangular trigonometry, exponential functions and statistical inference. Here, we are going to discuss the circle and its features.

Circle

It is the locus of all the points which is equidistant from the given point. That given point is also known as the centre of the circle.

There are some properties of a circle, as discussed below: –

• Centre of Circle

It is a point inside the circle which is equidistant from all the points on the circle.

• Radius of Circle

The radius of the circle is the distance from the centre to any point on the circle. It is half of the diameter of the circle.

Radius (r) =

Here, D is the diameter of a circle.

• Diameter of Circle

Diameter is double of circle radius. It is the length of a chord which passes through a centre of the circle.

Diameter = 2 x radius

One can easily calculate the radius of a circle with the use of the above formula.

If the diameter of a circle is known, one can easily calculate the radius of the circle.

• Circumference of Circle

It is the distance around the circle. Circumference of a circle is calculated with the use of the following formula.

C = 2 πr

Here, “C” is denoted as the circumference of the circle and   has a constant value that is 3.142 and “r” is the radius of the circle.

• Area of Circle

It is the region which is enclosed inside the circle. One can calculate the area of a circle with the use of following formula.

Radius =

A= (Radius)²

Here, “A” is the area of the circle and “ has a constant value that is 3.142.

• Chord of Circle

A line segment which joins any two points on the circle is known as a chord of a circle. One can find out the length of a chord with the use of formula given below when radius and central angle is given.

Chord length = 2r sin ()

Here, “r” is the radius of a circle, “c” is the angle which is subtended at the centre by chord and “sin” is the sine function.

And, when radius is given, then one can find out chord length with the use of following formulae.

Chord length = 2

• Tangent of Circle

The line which touches only one point of a circle is known as the tangent line of a circle. The equation of tangent is given as,

(y – y₀) = m_tgt (x – x₀)

• Secant

It is the line which intersects the circle at two distinct points in called secant. If the secant and a tangent of a circle are drawn from an outside point of the circle, then it is calculated by following formulae that is,

Length of secant x its external segment = (length of tangent segment)²

Some properties of the circle are discussed below: –

• If two circles have an equal radius, then the circles are said to be congruent.
• The most extended chord in the circle is known as diameter.
• A circle can circumscribe triangle, trapezium, square, kite and rectangle.
• Inside of square, kite and triangle, a circle can be inscribed.
• The chords which are equidistant from the centre of the circle are equal in length.
• When the length of chord increases, the perpendicular distance from the centre of the circle decreases.
• If tangents are drawn at the end of diameter, then they are parallel to each other.

Quiz

1. Find the area of a circle. If the radius of the circle is 3 inches?
2. 28 inches
3. 29 inches
4. 26 inches
5. 26 inches

Answer: – Option (c)

Explanation: –

As we know that area of a circle is calculated with the use of formula,

A= (Radius)²

A = 3.14 (3) (3)

A = 28.26 inches       

2. Find the area of a circle. If the diameter of a circle is given as 8 centimetres.
3. 24 cm
4. 50 cm
5. 51 cm
6. 24 cm

Answer: – Option (a)

Explanation: –

It is given that, d = 8 cm

we know that formula for calculation of diameter is, d = 2*r

Therefore, 8 = 2*r

r = 8 / 2

r = 4 cm

As now, we know the radius of the circle; therefore, one can easily calculate the area of a circle with the use of formula.

A =                         

A = 3.14 (4) (4)

A = 50.24 cm

3. If the area of circle is given as 5 cm². Find out radius of circle?
4. 5 cm
5. 10 cm
6. 7 cm
7. 6 cm

Answer: – Option (a)

Explanation: –

We know that, area of circle is find out with use of formula,

A =       

It is given that, A = 78.5 cm²                                                                        

Therefore, 78.5 = 3.14 * r²

78.5 / 3.14 = r²                                                                                               

r² = 25

r =

r = 5 cm

4. In the given figure, AC and BC are the radii of circles and the length of AB = 8 cm, AC = 4 cm. Therefore, find out the value of BC? (Note: – In the figure, BC is tangent to circle with centre A)
6. 4
7. 6
8. 4

Answer: – Option(b)

Explanation: –

As, ΔABC is a right-angled triangle.

Therefore, BC =

• 4

5. What is a formula for finding out the diameter of a circle?
6. d = 2*r
7. d = πr
8. d=2πr
9. d =

Answer:- Option(a)

Explanation:-

d = 2*r is the correct answer because the diameter is a double radius.

6. The diameter of the circle is given as 6 inches, and the radius of the circle is given as 12 inches. Determine if this is a True statement or False statement?
7. True
8. False

Answer: – Option (b)

Explanation: –

It is a false statement because the radius of a circle cannot be double or greater than the diameter of a circle.

7. What is the area of a circle? If the radius of the circle is given as 4 units?
8. 4 π
9. 8 π
10. 16 π
11. 20 π

Answer: – Option (c)

Explanation: –

we know that formula used for calculation of area is,

A = πR²

Here, Radius of circle is given as 4 units.

Therefore, A = π (4)²

A = π (4)²

A = 16 π

8. Find the area of two overlapping circles with radius 1 unit?
9. 2 π
10. π
11. 4 π
12. 6 π

Answer: – Option(b)

Explanation: –

Firstly, one has to find out the total area of both circles, we get

π+ π è 2 π

since both, the circles are overlapping. So, we cannot consider as twice, and we have to subtract.

Therefore, we get

2π – π = π

9. Find the area of the circle whose diameter is 8 inches?
10. 50 square inches
11. 40 square inches
12. 24 square inches
13. 24 square inches

Answer: – Option(a)

Explanation: –

We know that formula used for finding the area of the circle is, A = πr²

Now, one has to change the diameter to the radius, we get

r = ½ d

r = ½* 8

r = 4

Therefore, A = π * 4²

A = 16 π           (π   è 3.14)

A = 16 * 3.14

A = 50.24 square inches.

10. Find area of circle (in square inches) whose radius is 5 inches?
11. 78
12. 5
13. 6
14. 9

Answer: – Option(b)

Explanation: –

We know that, formula used for finding area of circle is, A = πr²

Radius = 5

Therefore, A = π * 5²

A = 25 π  (As, π   è 3.14)

A = 25 * 3.14

A = 78.5 square inches.