Pythagoras Theorem: Pythagoras’ Theorem is regarded as an important concept in Maths that finds immense applications in our day-to-day life. The theorem is named after the Greek mathematician and philosopher Pythagoras which explains the relationship between the three sides of a right-angled triangle. In this article, we will be providing you with all the necessary information about Pythagoras’ Theorem – statement, explanation, formula, proof, and examples.
Pythagoras Theorem: Everything You Need To Know
If one of the three angles of a triangle measures 90°, then we call it a right-angled triangle. The side of the triangle opposite to the right angle is called the hypotenuse of the triangle whereas the other two sides are called base and height respectively. Pythagoras’ Theorem explains the relationship between the hypotenuse, the base, and the height of a right-angled triangle.
Statement Of Pythagoras Theorem: Pythagoras Theorem Statement
The 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.”
This means that the area of the square on the hypotenuse of a right-angled triangle is equal to the sum of areas of the squares on the other two sides of the triangle.
Pythagoras’ Theorem Explanation
Check out the image below:
△ABC is a right-angled triangle with ∠B as the right angle.
AB is the height or altitude of the triangle.
Length of AB = a.
BC is the base of the triangle.
Length of BC = b.
AC, the side opposite to the right angle is the hypotenuse of the triangle.
Length of AC = c.
Let us now imagine three squares on the three sides of the triangle.
Now, as per the theorem, the area of the square on the hypotenuse of the triangle is equal to the sum of the areas of the squares on the other two sides.
Pythagoras Theorem Formula
Referring to the above image, the theorem can be expressed as:
(Hypotenuse)2 = (Height)2 + (Base)2 or c2 = a2 + b2
Pythagoras Theorem Proof
The proof of Pythagorean Theorem is provided below:
Let us consider the right-angled triangle △ABC wherein ∠B is the right angle (refer to image 1).
In order to prove (AB)2 + (BC)2 = (AC)2, let’s draw a perpendicular line from the vertex B (bearing the right angle) to the side opposite to it, AC (the hypotenuse), i.e. BO ⊥ AC.
Now, in △ABC and △ABO, we have:
- i) ∠A is common.
- ii) ∠AOB = ∠ABC (Both are 90°)
Therefore, △ABC ~ △ABO (By AA-similarity)
So, AO/AB = AB/AC.
=> (AB)2 = AO × AC ——– (1)
Now, in △ABC and △OBC, we have:
- i) ∠C is common.
- ii) ∠BOC = ∠ABC (Both are 90°)
Therefore, △ABC ~ △OBC (By AA-similarity)
So, OC/BC = BC/AC.
=> (BC)2 = OC × AC ——– (2)
Adding equation 1 and equation 2, we have:
(AB)2 + (BC)2 = AO × AC + OC × AC
=> (AB)2 + (BC)2 = AC (AO + OC)
=> (AB)2 + (BC)2 = AC × AC (Now, since AO + OC = AC)
=> (AB)2 + (BC)2 = (AC)2
Hence, the Pythagorean Theorem is proved.
Applications Of Pythagorean Theorem
Apart from solving various mathematical problems, Pythagorean Theorem finds applications in our day-to-day life as well, such as, in:
- i) Architecture and construction, let’s say to construct a square corner between two walls, to construct roofs, etc.
- ii) Cartography
- iii) Surveying
- iv) Painting and design
- v) Navigation
Examples Of Pythagoras Theorem
Some example problems related to Pythagorean Theorem are as under:
Example 1: The length of the base and the hypotenuse of a triangle are 6 units and 10 units respectively. Find the length of the third side (height).
Length of base = 6 units
Length of hypotenuse = 10 units
We know that (Hypotenuse)2 = (Height)2 + (Base)2 .
=> (10)2 = (Height)2 + (6)2
=> 100 = (Height)2 + 36
=> (Height)2 = 100 – 36
=> (Height)2 = 64
Therefore, Height = √(64) = 8 units.
Example 2: The length and breadth of a rectangle are 5 units and 11 units respectively. Find the length of the diagonal.
Each of the four angles of a rectangle measures 90°.
Therefore, each diagonal divides the rectangle into two right-angled triangles, with the diagonal being the hypotenuse of each of the triangles and the length and breadth being the other two sides.
As (Hypotenuse)2 = (Height)2 + (Base)2,
(Hypotenuse)2 = (5)2 + (11)2 = 25 + 121 = 146
Therefore, Hypotenuse (Diagonal of the Rectangle) = √(146) = 12.083 units.
So, now you know everything about Pythagoras Theorem. Solve more questions of varying types and master the concept.
Pythagoras Theorem: Important FAQs
Some of the important FAQs related to the Pythagoras Theorem are:
Q1: What is Pythagorean theorem?
Ans: 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.
Q2: Who invented the Pythagorean theorem?
Ans: Pythagorean theorem is named after the Greek mathematician Pythagoras.
Q3: How can I prove the Pythagorean Theorem?
Ans: You can prove the Pythagorean Theorem in three ways:
– Using Coordinate Geometry
– Using Trigonometry
– Using Similarity
This article contains proof of the Pythagorean Theorem using the triangle similarity method.
Q4: What are some real-life uses of the Pythagorean theorem?
Ans: Pythagoras Theorem has a lot of real life uses. Construction workers, Architects, Carpenters, Framers, etc. all use its concepts.
Q5: Write the statement of Pythagoras Theorem.
Ans: The statement of Pythagoras Theorem is: “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides.”
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