1) Information about Pythagoras theorem.., definition, Formula
2) Rules of Pythagoras theorem.
3) Examples of Pythagoras theorem..
4) Use of Pythagoras theorem..
Answers
Answer:
If we consider what the distance formula really tells you, we can see the similarities. It is more than just a similar form.
The distance formula is commonly seen as:
D
=
√
(
x
1
−
x
2
)
2
+
(
y
1
−
y
2
)
2
We commonly write the Pythagorean Theorem as:
c
=
√
a
2
+
b
2
Consider the following major points (in Euclidean geometry on a Cartesian coordinate axis):
The definition of a distance from
x
to
±
c
is
|
x
−
c
|
.
There is the relationship where
√
(
x
−
c
)
2
=
|
x
−
c
|
=
x
−
c
AND
−
x
+
c
The distance from one point to another is the definition of a line segment.
Any diagonal line segment has an
x
component and a
y
component, due to the fact that a slope is
Δ
y
/
Δ
x
. The greater the
y
contribution, the steeper the slope. The greater the
x
contribution, the flatter the slope.
What do you see in these formulas? Have you ever tried drawing a triangle on a Cartesian coordinate system? If so, you should see that these are two formulas relating the diagonal distance on a right triangle that is composed of two component distances
x
and
y
.
Or, we could put it another way through substitutions based on the distance definitions above. Let:
x
1
−
x
2
=
±
a
y
1
−
y
2
=
±
b
(depending on if
x
1
>
x
2
or
x
1
<
x
2
, and similarly for
y
.)
Now what do you see? An equivalence.
D
=
√
(
±
a
)
2
+
(
±
b
)
2
=
c
=
√
a
2
+
b
2
In short, the distance formula is a formalization of the Pythagorean Theorem using
x
and
y
coordinates. In other words, they are the same thing in two seemingly different contexts.
Pythagoras Theorem is an important topic in Maths, which explains the relation between the sides of a right-angled triangle. It is also sometimes called the Pythagorean Theorem. The formula and proof of this theorem are explained here with examples.
Pythagoras theorem is basically used to find the length of an unknown side and angle of a triangle. By this theorem, we can derive base, perpendicular and hypotenuse formula. Let us learn mathematics of Pythagorean theorem in detail here.
Table of Contents:
Statement
Formula
Proof
Applications
Problems
Pythagoras Theorem Statement
Pythagoras theorem states that “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides“. The sides of this triangle have been named as Perpendicular, Base and Hypotenuse. Here, the hypotenuse is the longest side, as it is opposite to the angle 90°. The sides of a right triangle (say a, b and c) which have positive integer values, when squared, are put into an equation, also called a Pythagorean triple.
Pythagoras Theorem-Right Angle Triangle
History
The theorem is named after a greek Mathematician called Pythagoras.
Pythagoras Theorem Formula
Consider the triangle given above:
Where “a” is the perpendicular side,
“b” is the base,
“c” is the hypotenuse side.
According to the definition, the Pythagoras Theorem formula is given as:
Hypotenuse2 = Perpendicular2 + Base2
c2 = a2 + b2
The side opposite to the right angle (90°) is the longest side (known as Hypotenuse) because the side opposite to the greatest angle is the longest.
Pythagoras Theorem
Consider three squares of sides a, b, c mounted on the three sides of a triangle having the same sides as shown.
By Pythagoras Theorem –
Area of square A + Area of square B = Area of square C
Example
The examples of theorem based on the statement given for right triangles is given below:
Consider a right triangle, given below:
Pythagoras theorem example
Find the value of x.
X is the side opposite to right angle, hence it is a hypotenuse.
Now, by the theorem we know;
Hypotenuse2 = Base2 + Perpendicular2
x2 = 82 + 62
x2 = 64+36 = 100
x = √100 = 10
Therefore, we found the value of hypotenuse here.
Right Angle Triangle Theorem
Types Of Triangles
Triangles Class 9
Triangles For Class 10
Class 10 Maths
Important Questions Class 10 Maths Chapter 6 Triangles
Pythagoras Theorem Proof
Given: A right-angled triangle ABC, right-angled at B.
To Prove- AC2 = AB2 + BC2
Construction: Draw a perpendicular BD meeting AC at D.
Pythagoras theorem Proof
Proof:
We know, △ADB ~ △ABC
Therefore, \frac{AD}{AB}=\frac{AB}{AC} (corresponding sides of similar triangles)
Or, AB2 = AD × AC
Also, △BDC ~△ABC
Therefore, \frac{CD}{BC}=\frac{BC}{AC} (corresponding sides of similar triangles)
Or, BC2= CD × AC
Adding the equations (1) and (2) we get,
AB2 + BC2 = AD × AC + CD × AC
AB2 + BC2 = AC (AD + CD)
Since, AD + CD = AC
Therefore, AC2 = AB2 + BC2
Hence, the Pythagorean theorem is proved.
Note: Pythagorean theorem is only applicable to Right-Angled triangle.
39,41,011
Applications of Pythagoras Theorem
To know if the triangle is a right-angled triangle or not.
In a right-angled triangle, we can calculate the length of any side if the other two sides are given.
To find the diagonal of a square.
Useful For
Pythagoras theorem is useful to find the sides of a right-angled triangle. If we know the two sides of a right triangle, then we can find the third side.
How to use?
To use this theorem, remember the formula given below:
c2 = a2 + b2
Where a, b and c are the sides of the right triangle.
For example, if the value of a = 3 cm, b = 4 cm, then find the value of c.
We know,
c2 = a2 + b2
c2 = 32+42
c2 = 9+16
c2 = 25
c = √25
c = 5
Hence, the third side is 5 cm.
As we can see, a + b > c
3 + 4 > 5
7 > 5
Hence, c = 5 cm is the hypotenuse of the given triangle.
How to find whether a triangle is a right-angled triangle?
If we are provided with the length of three sides of a triangle, then to find whether the triangle is a right-angled triangle or not, we need to use the Pythagorean theorem.
Let us understand this statement with the help of an example.
Suppose a triangle with sides 10, 24, and 26 are given.
Clearly, 26 is the longest side.
It also satisfies the condition, 10 + 24 > 26
We know,
c2 = a2 + b2
So, let a = 10,b = 24 and c = 26
First we will solve R.H.S. of equation 1.
a2 + b2 = 102 + 242 = 100 + 576 = 676
Now, taking L.H.S, we get;
c2 = 262 = 676
We can see,
LHS = RHS
Therefore, the given triangle is a right triangle, as it satisfies the theorem.