Isosceles Triangle Proof

In an isosceles triangle, two sides are equal in length. An isosceles triangle also has two angles of the same measure; namely, the angles opposite to the two sides of the same length; this fact is the content of the Isosceles triangle theorem. Some mathematicians define isosceles triangles to have only two equal sides, whereas others define that an isosceles triangle is one with at least two equal sides. The latter definition would make all equilateral triangles isosceles triangles. (Source : WIKIPEDIA)

Things need to remember for Proofs of isosceles triangle:

For proving isosceles triangle we need to know the following information about  isosceles triangle,
There are two types of isosceles triangles,
1.Normal isosceles triangle
2.Right isosceles triangle.

Properties:
Properties of parts of  isosceles triangle:
1.The two sides of the isosceles tringles are equal.
2.Two base angles has same measure.
3. Angle ratio of the right isosceless triangle is   45:90:45.
4.The side ratio of the isosceles triangle is  1:1:`sqrt(2)`

Problems on isosceles triangle proof:

Problem 1:
Prove that the following triangle is isosceles triangle.

Proof:
Given , The angle We know that the sum of the angles are 180.
So 100 + x +x = 180
      100 + 2x = 180
Subtract 100 0n both sides.
    2x =180 -100
    2x = 80
Divide by x on both sides,
 x = 40
The base anles are equal two 40 .
According to the properties of isosceles triangle,we can determine that the given triangle is isosceles triangle.
Hence the proof.

Problem 2:
Prove that the triangle with the sides 5 : 5 : 5`sqrt(2)` is an isosceles right triangle triangle.
Proof:
Given,The sides of the triangle is 5 , 5 ,5`sqrt(2)`
We know that in a right Angle triangle,the hypotenuse is greater then the legs and it satisfies the pythagorean theorem,
5`sqrt(2)` > 5 , 5
5²+5² = (5`sqrt(2)` )²
25+25 = 25 *2
50 = 50
So the given sides satisfies the pythagoren theorem.So we can say that it is a right triangle.
The given sides are in the ratio of 1:1:`sqrt(2)`
so the given triangle is isosceles triangle.
Hence the proof.

Pyramid with a Trapezoid Base

A pyramid is a building where the outer surfaces are triangular and converge at a point. The base of a pyramid can be trilateral, quadrilateral, or any polygon shape, meaning that a pyramid has at least three outer surfaces (at least four faces including the base). The square pyramid, with square base and four triangular outer surfaces, is a common version. (Source: Wikipedia)

I like to share this Trapezoid Shape with you all through my article.

Pyramid with trapezoid base:
      A pyramid with a trapezoid base is called as trapezoidal pyramid.
     Volume of trapezoidal pyramid = (1 / 3) * ((1 / 2) * (base₁ + base₂) * height

Example problems for pyramid with a trapezoid base

Trapezoid pyramid problem 1:
        A base length of the trapezoid pyramid is 12 cm, 14 cm and its height of the pyramid is 6 cm. Find the volume of the trapezoidal pyramid.
Solution:
     Given base lengths are b₁ = 12 cm, b₂ = 14 cm, and h = 6 cm.
Formula:
   Volume of trapezoidal pyramid = (1 / 3) * ((1 / 2) * (base₁ + base₂) * height
Substitute the given values in the above formula, we get
= (1 / 3) * ((1 / 2) * (12 + 14) * 6 cm³
= (1 / 3) * 78 cm³
= 26 cm³

Answer:
 Volume of the trapezoidal pyramid is 26 cm³

Trapezoid pyramid problem 2:
        A base length of the trapezoid pyramid is 15 cm, 24 cm and its height of the pyramid is 9 cm. Find the volume of the trapezoidal pyramid.
Solution:
     Given base lengths are b₁ = 15 cm, b₂ = 24 cm, and h = 9 cm.
Formula:
Volume of trapezoidal pyramid = (1 / 3) * ((1 / 2) * (base₁ + base₂) * height
 Substitute the given values in the above formula, we get
= (1 / 3) * ((1 / 2) * (15 + 24) * 9 cm³
 = (1 / 3) * 175.5 cm³
                                               = 58.5 cm³
Answer:
 Volume of the trapezoidal pyramid is 58.5 cm³

Trapezoid pyramid problem 3:
        A base length of the trapezoid pyramid is 6 cm, 11 cm and its height of the pyramid is 5 cm. Find the area of the trapezoid.
Solution:
     Given base lengths are b₁ = 6 cm, b₂ = 11 cm, and h = 5 cm.
Formula:
Area of trapezoid = ((1 / 2) * (base₁ + base₂) * height
Substitute the given values in the above formula, we get
                                               = (1 / 2) * (6 + 11) * 5 cm³
                                               = (1 / 2) * 85 cm³
                                               = 42.5 cm²
Answer:
 Volume of the trapezoidal pyramid is 42.5 cm²

Practice problems for pyramid with a trapezoid base

Trapezoid pyramid problem 1:
        A base length of the trapezoid pyramid is 7 cm, 9 cm and its height of the pyramid is 12 cm. Find the volume of the trapezoidal pyramid.
Answer:
 Volume of the trapezoid pyramid is 32 cm³
Trapezoid pyramid problem 2:
        A base length of the trapezoid pyramid is 10 cm, 22 cm and its height of the pyramid is 7 cm. Find the volume of the trapezoidal pyramid.
Answer:

 Volume of the trapezoid pyramid is 37.33 cm³