Construction of Triangles

Introduction to construction of triangles:

The triangles can be constructed if the following requirements are given such as follows,

The measurement of three sides should be given (or)

The measurement of the two sides and the included angle should be given (or)

The measurement of a side and any two angles should be given.

Now we are going to see about the construction of triangles. Is this topic Scalene Triangles hard for you? Watch out for my coming posts.


Construction of triangles:


Construction of triangles if three sides are given:

Construct a triangle if three sides are given with x, y and z measurements.

Steps of construction:

First a line segment QR of x cm length should be drawn.

With Q as center and radius of y cm be drawn and it equals to PQ and draw an arc of a circle.

With R as center and radius of PR = z cm and draw an arc and it will intersects at the first arc of point P.

Now join the points of line segments PQ and PR.

Thus, PQR is a required triangle. I have recently faced lot of problem while learning Geometry Definition, But thank to online resources of math which helped me to learn myself easily on net.

Other constructions of triangles:

Construction of triangles if two sides and angle are given:

Construct a triangle if two sides and an angle are given.

Steps of Construction:

First we have to draw a ray of QX of some length.

With the help of protractor measure the given angle and draw the line to meet Q.

The ray QY which may cut line segment QR of x cm.

The ray QY which may cut the line segment QP of y cm.

Now we can join the two points P and R.

Thus, PQR is the required triangle.

Construction of triangles if two angles and Side are given:

Construct a triangle if two angles and a side are given.


Steps of Construction:

First we should draw the line segment of QR of given length.

With the help of the protractor measure the given angle at RQX

Then, draw QRY for the given angle such that XY lie on the same side of the PQ.

Then, label the point where it intersects at QX and QY as P.

Thus, the PQR is the required triangle.

Base Pentagon Prism

Introduction:-
In geometry, the pentagonal prism is a prism with a pentagonal base.

Pentagonal prism is a type of heptahedron.

If faces are all regular, the pentagonal prism is a semi regular polyhedron and the third in an infinite set of prisms formed by square sides and two regular polygon caps.

A pentagonal prism has                                                                                                                                                                         Source:- Wikipedia.

7 faces
10 vertices
15 edges.
The pentagonal prism looks like .


Formulas For Pentagonal Prism:-


`Base area = 5/ 2 * a * s`
`Base perimeter = 5s`
`Prisms Surface area = 5as + 5sh.`
`Volume of Prism = (5/2)ash`
`here`
`a = apothem Leng th,`
`s = side,`

`h = height.`

Please express your views of this topic Congruence of Triangles by commenting on blog.

Problems on Prism:-


Problem 1:-

Find the base area  and base perimeter of the pentagonal prism if the apothem length is 4 and the side is 2.

Solution:-

Given

The apothem, length is 4

The side is 2.

The formula that is used to calculate the area of the base is `5/ 2 a* s.`

By plugging in the values of  a and s in the formula we get

Area of the base = `5/ 2` * 4 * 2.

By crossing 2 and 2 in the above equation we get

Area of the base = 5 * 4 = 20.square units.

The formula that is used to find the base perimeter is  `5s` .

By plugging in the given values we get

Base perimeter = 5 * s = 5 * 2 = 10.

Problem 2:-

Find the base area  and base perimeter of the pentagonal prism if the apothem length is 6 and the side is 4.

Solution:-

Given

The apothem, length is 6

The side is 4.

The formula that is used to calculate the area of the base is` 5/ 2 a* s.`

By plugging in the values of  a and s in the formula we get

Area of the base = `5/ 2 * 6 * 4.`

By crossing 2 and 4 in the above equation we get

Area of the base = 5 * 12 = 60.square units.

The formula that is used to find the base perimeter is  `5s.`

By plugging in the given values we get

Base perimeter = 5 * s = 5 * 4 = 20.

Problem 3:-

Find the base area and base perimeter of the pentagonal prism if the apothem length is 5 and the side is .

Solution:-

Given

The apothem, length is 5

The side is 7.

The formula that is used to calculate the area of the base is `5/ 2 a* s.`

By plugging in the values of  a and s in the formula we get

Area of the base = `5/ 2 * 5 * 7.`

Area of the base = `175/ 2` square units.

The formula that is used to find the base perimeter is  5s.

By plugging in the given values we get

Base perimeter = 5 * s = 5 * 7 = 35.

angles at a point

Introduction (Angles at point):

In geometry an angle is the figure produced by two ray’s distribution a common endpoint, called the vertex of angle. The degree of the angle is the quantity of revolution that separates the two waves, and deliberate by considering the length of circular curve is out when one ray is rotate regarding the vertex to correspond with the other. The angle along with a line and a curve or along with two intersecting curve.


Positive and negative angles at a point:

In mathematical script is that angles specified a sign are positive angles if considered anticlockwise and negative angles ? is efficiently the same to a positive angle of one full rotation less ?. if considered clockwise, from a known line. If no line is specified, that can be understood to be the x-axis in the Cartesian plane. In many geometrical situations a pessimistic angle of ?? is efficiently the same to a positive angle of one full rotation less ?.

Example, a clockwise rotation of 45° (angle of ?45°) is efficiently the same to an anticlockwise rotation of 360° ? 45° (angle of 315°).

Types of Angles:

Right angle
Acute angle
obtuse angle
reflex angle
Vertical opposite angles
Co-responding angles and Alternative angles
Interior angle
Identifying angles:

Angles may be recognized by the labels involved to the three points to identify them. Example, the angle by vertex A with this by the rays AB and AC.

Potentially, an angle denoted,  ?BAC may refer to any of four angles: the clockwise angle from B to C, the anticlockwise angle from B to C, the clockwise angle as of C to B, or the anticlockwise angle as of C to B, wherever the way in that the angle is deliberate determines its sign.


Examples for angles at a point:


Example 1:

Find the value of x.

Solution:

x + 80° + 2x + x = 180° (contiguous angles on a straight line)

4x = 180° - 80°
= 100°

x = 100°
4
The answer of x = 25°

Example 2:

Find the value of x.

Solution:

48° + 90° + 120° + x = 360° ( Angles at a point )
x = 360° - (48° + 90° + 120° )
= 360° - 258°
The answer of x= 102°