Semicircle Learning

Introduction of semicircle learning :

Semicircle is defined as half of a circle. That is, the angle is 180 degree arc of a circle. A triangle decorated in a semicircle is always called a right triangle.

If two curves or arcs are equal, then both the segments and sectors are similar. This each part of term is called as semicircle region.

Formulas of Semicircle Learning

A semicircle is the area enclosed by a diameter and an arc of the circle joining its two ends. The length of the resulting segment is called the geometric mean, which can be proved using the concept of Pythagorean Theorem.

Formulas:

Area of semicircle (A) =circle /2

A = (pr2)/2

Circumference of semicircle(C) = (2pr)/2

C = pr

A circumference of a semicircle is calculated for the circumference of circle divided by 2.we get,

C = 2pr ==> C/2 = pr

p = 3.14 ( approximately )

A perimeter of a semicircle is the sum of circumference and diameter of a semicircle. We get,

P = pr + 2r = r (p+2)

P = 5.14 r

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Examples of Semicircle Learning:

Semicircle learning Ex 1!:

Find the area of semicircle with radius of 12.5 cm.

Semicircle learning sol :

We can find the area of semicircle by using the following formula,

Area = (pr2)/2

Substitute the values of p and the radius into the above formula. Then we get,

= (3.14*(12.5)2)/2

Squaring the values of radius and multiplying with 3.14 then dividing by the value of 2.

= (3.14*156.25)/2

= (490.625)/2

Then we get the final answer.

=245.3 cm2

Answer: 245.3 cm2

Semicircle learning Ex 2:

Find the perimeter of semicircle with the radius 10 cm.

Semicircle learning sol :

We can find the perimeter of semicircle by using the following formula,

Perimeter = 5.14*r

Substitute the value of r into the above formula,

=5.14*10

=51.4 cm

Answer: 51.4 cm

Semicircle learning Ex 3:

Find the circumference of semicircle with the radius of 7.5 cm.

Semicircle learning sol :

We can find the circumference of semicircle by using the following formula,

Circumference C = (2pr)/2

Circumference C = pr

Substitute the value of p and the radius.

C = 3.14*7.5

C = 23.55 cm2

Answer: C = 23.55 cm2

Measuring Irregular Triangles

Introduction for measuring irregular triangles:

The irregular triangle is nothing but the triangle where the three sides are not equal and the angles present in it also different during its measurement. The only irregular triangle is the scalene triangle. Now we are going to see about the measuring of irregular triangle with some example problems.


About Measuring of Irregular Triangle:
Now we are going to see about the irregular triangles and its measurement. The irregular triangle is nothing but the scalene triangle where the sides are unequal in its length and the angles in it also unequal.

When the two sides and an angle are given and if we want to find the third side of an irregular triangle, then use the formula which is given below,

c2 = a2 + b2 - 2ac * cos ?

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Problems for Measuring Irregular Triangle:

Example 1:

The sides of the triangle are 5cm and 10 cm and the angle measuring is 40. Determine the third side of the irregular triangle.

Solution:

The third side of the triangle can be calculated by using the formula,

c2 = a2 + b2 - 2ac * cos ?

Now substitute the values in the formula we get,

c2 = 52 + 102  2(5)(10) * cos 40

Now square the values which are substituted in the formula as follows,

= 25 + 100  100 cos 40

c2 = 48.39

c = `sqrt 48.39`

c = 6.95

The value can be rounded as 7cm.

Example 2:

Find the third side of the triangle whose measurements of the triangle are 7cm, 8cm and angle measuring is about 50 degree.

Solution:

The third side of the triangle can be calculated by using the formula,

c2 = a2 + b2 - 2ac * cos ?

Now substitute the values in the formula we get,

c2 = 72 + 82  2(7)(8) * cos 40

= 49 + 64  112 cos 50

c2 = 41

c = ` sqrt 41`

c = 6.4

The value can be rounded as 6cm.

Non Coplanar Definition

Introduction to non-coplanar points:
The points which do not lie in the same plane or geometrical plane are called as non-coplanar points. Any 3 points can be enclosed by one plane or geometrical plane but four or more points cannot be enclosed by one. The points belong to the same plane are called as coplanar points. In this article we shall be discussing the non-coplanar points. Now we know what non-coplanar point is and we shall see some examples of the non-coplanar points and solve it for the same.

Example for Non Co-planar Points
1)      From the below shown figure the points are non coplanar points as they do not lie on the same plane it lies in different planes.

2)      We can see four planes with the help of four non co-planar points.

3)      Plane is the two dimensional geometrical object.

Non co-planar points

Non co-planar points

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Solved Examples for Non Co-planar Points:

Ex 1:  Check whether the following lines are co-planar or not.

3x+6y+9 = 0 and 4x+4y+11 = 0

Sol :  The given equations are  3x+6y+9 = 0 and 4x+4y+11 = 0

The slope intercept form can be given as y = mx+b

Where m indicates slope.

Comparing the above equation with the given equation, we get:

6y = -2x-9

Dividing by 6 on both sides we get:

We get,

--- (1)

The slope intercept form can be given as y = nx+b

Where n indicates slope.

Comparing the above equation with the given equation, we get:

4y = -4x-11

Dividing by 4 on both sides we get:

y = -1

We get n = -1--------- (2)

Equation (1) (2), that is m n

That is the slopes of the two equations are not equal and therefore the points lie on the two lines are non co-planar points.

Ex 2:   Check whether the following lines are co-planar or not.

x+5y+9 = 0 and 2x+10y+11 = 0

Sol :  The given equations are  x+5y+9 = 0 and 2x+10y+11 = 0

The slope intercept form can be given as y = mx+b

Where m indicates slope.

Comparing the above equation with the given equation, we get:

5y = -x-9

Dividing by 5 on both sides we get:

We get,

------- (1)

The slope intercept form can be given as y = nx+b

Where n indicates slope.

Comparing the above equation with the given equation, we get:

10y = -2x-11

Dividing by 10 on both sides we get:

We get

--------- (2)

Equation (1) =(2), that is m = n

That is the slopes of the two equations are equal and therefore the points lie on the two lines are co-planar points.